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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span></span><br/>

<a NAME="E1:70"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
;<br/>
<a NAME="E2:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> 
 
;<br/>
<a NAME="E3:70"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E4:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num2.html#T51" target="_self" title="FIB_NUM2:th.51">Th53</a>, <a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span>)</span> = 
<a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E6:70"/>
<span class="p1">(<span class="default"><a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fib_num2.html#K3" title="FIB_NUM2:func.3">EvenNAT</a> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</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"> verum</span><br/>


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