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

<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/>

<a NAME="E1:21"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> iff <font color="Olive" title="b1">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> iff <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:21_1"/>
( <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> implies <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> implies <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1"/>
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E3:21_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/>
<a NAME="E4:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
;<br/>
<a NAME="E6:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E7:21_1_1"/>
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="funct_3.html#T2" target="_self" title="FUNCT_3:th.2">Th2</a></span>;<br/>
<a NAME="E8:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<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>
 
;<br/>
<a NAME="E9:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E10:21_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21_1_1"/>
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:21_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E5:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_3.html#T7" target="_self" title="FUNCT_3:th.7">Th8</a></span>;<br/>
<a NAME="E6:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<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="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E7:21_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E8:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="funct_3.html#T3" target="_self" title="FUNCT_3:th.3">Th3</a></span>;<br/>
<a NAME="E9:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E10:21_1"/>
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E3:21"/>
 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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<a NAME="E3:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T2" target="_self" title="FUNCT_3:th.2">Th2</a></span>;<br/>
<a NAME="E4:21_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/>
<a NAME="E5:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E6:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
;<br/>
<a NAME="E7:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<font color="Maroon" title="c3">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_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="funct_3.html#T7" target="_self" title="FUNCT_3:th.7">Th8</a></span>
<br/>.= 
<font color="Maroon" title="c2">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T126" title="RELAT_1:th.126">RELAT_1:126</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="funct_3.html#D1" target="_self" title="FUNCT_3:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:21"/>
 <a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
