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

<a NAME="E1:2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E12"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E2:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A1</font></span>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E18"><span class="lab"><font color="Green" title="E7">Lm7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E12"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E4:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E17"><span class="lab"><font color="Green" title="E6">Lm6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E12"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E6:2"/><span class="kw">then </span>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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">1,1 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E16"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a href="realset2.html#K1" title="REALSET2:func.1">add_2</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> 1</span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> 1</span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E12"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="funcop_1.html#T86" title="FUNCOP_1:th.86">FUNCOP_1:86</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
