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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<div><i><font color="Green" title="E1">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c4">x</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"><span class="p2">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 ;<br/><a NAME="E2:4_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E3:4_1_1"/><b>then </b>
<font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:4_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c5">y1</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> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E6:4_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c6">z1</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> <font color="Maroon" title="c2">g</font>
 <b>and </b><br/><a NAME="E7:4_1_1"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c4">x</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="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>take </b><font color="Maroon" title="c7">z1</font> = <font color="Maroon" title="c6">z1</font>;<br/><b>take </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y1</font>;<br/><b>thus </b><a NAME="E8:4_1_1"/>
<font color="Maroon" title="c4">x</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="c8">y1</font>,<font color="Maroon" title="c7">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E7:4_1_1"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/><a NAME="E9:4_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E10:4_1_1"/><b>then </b><i><font color="Green" title="E6">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">y1</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E5:4_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="altcat_2.html#E6:4_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:4_1_1"/><i><font color="Green" title="E7">A7</font></i>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c7">z1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E5:4_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="altcat_2.html#E6:4_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E12:4_1_1"/><i><font color="Green" title="E8">A8</font></i>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">y1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E5:4_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="altcat_2.html#E6:4_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E13:4_1_1"/>
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c7">z1</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"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:4_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="altcat_2.html#E7:4_1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E14:4_1_1"/><b>then </b>
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">y1</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> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E11:4_1_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="txt" href="altcat_2.html#E12:4_1_1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="funct_4.html#T43" title="FUNCT_4:th.43">FUNCT_4:43</a></i>;<br/><b>hence </b><a NAME="E15:4_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">y1</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E10:4_1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:4_1"/><i><font color="Green" title="E2">A9</font></i>: 
<font color="Maroon" title="c4">x</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="c6">z</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:4_1"/><i><font color="Green" title="E3">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 ;<br/><a NAME="E5:4_1"/><i><font color="Green" title="E4">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E4:4_1"><i><font color="Green" title="E3">A10</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E7:4_1"/><b>then </b><i><font color="Green" title="E5">A12</font></i>: 
( <font color="Maroon" title="c5">y</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> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c6">z</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> <font color="Maroon" title="c3">h</font> )
 <b>by </b><i><a class="txt" href="altcat_2.html#E5:4_1"><i><font color="Green" title="E4">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E9:4_1"/><b>then </b><i><font color="Green" title="E6">A13</font></i>: 
<font color="Maroon" title="c4">x</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E3:4_1"><i><font color="Green" title="E2">A9</font></i></a>, <a class="txt" href="altcat_2.html#E7:4_1"><i><font color="Green" title="E5">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:4_1"/><i><font color="Green" title="E7">A14</font></i>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E7:4_1"><i><font color="Green" title="E5">A12</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E11:4_1"/><i><font color="Green" title="E8">A15</font></i>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E7:4_1"><i><font color="Green" title="E5">A12</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E12:4_1"/>
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</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> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E4:4_1"><i><font color="Green" title="E3">A10</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E13:4_1"/><b>then </b>
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E3:4_1"><i><font color="Green" title="E2">A9</font></i></a>, <a class="txt" href="altcat_2.html#E10:4_1"><i><font color="Green" title="E7">A14</font></i></a>, <a class="txt" href="altcat_2.html#E11:4_1"><i><font color="Green" title="E8">A15</font></i></a>, <a class="ref" href="funct_4.html#T43" title="FUNCT_4:th.43">FUNCT_4:43</a></i>;<br/><b>hence </b><a NAME="E14:4_1"/>
<font color="Maroon" title="c4">x</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"><span class="p2">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E9:4_1"><i><font color="Green" title="E6">A13</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green" title="E2">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 ;<br/><a NAME="E2:4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E3:4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E4:4_2"/><b>then </b><i><font color="Green" title="E3">A17</font></i>: 
( <font color="Maroon" title="c4">y</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> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c5">z</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> <font color="Maroon" title="c3">h</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:4_2"/>
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</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> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:4_2"><i><font color="Green" title="E2">A16</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:4_2"/><b>then </b><i><font color="Green" title="E4">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span></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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E4:4_2"><i><font color="Green" title="E3">A17</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E7:4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</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"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</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="altcat_2.html#E4:4_2"><i><font color="Green" title="E3">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</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="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:4_2"><i><font color="Green" title="E3">A17</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E6:4_2"><i><font color="Green" title="E4">A18</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:4_2"><i><font color="Green" title="E3">A17</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E1:4_2"><i><font color="Green" title="E2">A16</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>;<br/>


</div>
