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

<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>,<font color="Maroon" title="c1">X</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <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="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>,<font color="Maroon" title="c1">X</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>;<br/>
<a NAME="E1:7"/><span class="lab"><font color="Green" title="E7">A1</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#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/>
<a NAME="E2:7"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</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> 1 ) &amp; (  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies ( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:7_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <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"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:7_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:7_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/>
<a NAME="E4:7_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</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="E3:7_1"/>
 not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <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"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E4:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E5:7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></span>;<br/>
<a NAME="E6:7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</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="E3:7"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T62" title="FUNCT_4:th.62">FUNCT_4:62</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</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> <span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T39" title="FUNCT_3:th.39">FUNCT_3:39</a></span>;<br/>
<a NAME="E5:7"/><span class="kw">then </span>
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:7"/>
<span class="p1">(<span class="default">(<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>,<font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</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>
