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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:107"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:107"/><i><font color="Green">E72</font></i>: 
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:107"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="comput_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E5:107"/><i><font color="Green">E74</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> as  <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="ref" href="comput_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E7:107"/><i><font color="Green">E75</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:107_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:107_1"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E3:107_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E1:107_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><a NAME="E5:107_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:107_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E5:107"><i><font color="Green">E74</font></i></a>, <a class="txt" href="comput_1.html#E1:107_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E9:107_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:107_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:107_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E11:107_1"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E13:107_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:107_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:107_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E9:107_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E11:107_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="comput_1.html#T53" target="_self">Th53</a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><i><font color="Green">E83</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E15:107_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_1"><b>suppose </b></a><a NAME="E1:107_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E2:107_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E7:107_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E9:107_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E11:107_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E16:107_1"><i><font color="Green">E83</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_2"><b>suppose </b></a><a NAME="E1:107_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:107_1_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:107_1_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E16:107_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_1_2_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green">Lemma68</font></i></a></i>;<br/><b>hence </b><a NAME="E5:107_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E2:107_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E7:107_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E9:107_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E11:107_1"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_3"><b>suppose </b></a><a NAME="E1:107_1_2_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:107_1_2_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>thus </b><a NAME="E4:107_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E2:107_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E7:107_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E9:107_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E11:107_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E16:107_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_1_2_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E80"><i><font color="Green">Lemma69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0;<br/>
<a NAME="E9:107"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>now </b></a><div class="add"><a NAME="E1:107_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:107_2"/><i><font color="Green">E80</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:107_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:107_2"/><i><font color="Green">E82</font></i>: 
( (  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) implies <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E6:107_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self">Def13</a></i>;<br/><i><font color="Green">E83</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E9:107"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><b>thus </b><a NAME="E8:107_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span> iff  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E3:107_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E5:107_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E7:107_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>

<b>assume </b><a NAME="E2:107_2_2"/>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>
;<br/>
<b>hence </b><a NAME="E4:107_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E3:107_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E7:107_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E9:107_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span>
 ;<br/><a NAME="E10:107_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E3:107_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E4:107_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E5:107_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E7:107_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E11:107_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:107_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="comput_1.html#E3:107_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E7:107_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:107"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>thus </b><a NAME="E13:107"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> iff  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E10:107"><i><font color="Green">E78</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:107_3"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:107_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:107"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E4:107_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E10:107"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E2:107_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:107_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E5:107"><i><font color="Green">E74</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:107_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E3:107_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E11:107"><i><font color="Green">E79</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E4:107_4"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E10:107"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E1:107_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<a NAME="E15:107"/><i><font color="Green">E80</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E16:107"/><i><font color="Green">E81</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>;<br/>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_5"><b>now </b></a><div class="add"><a NAME="E1:107_5"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:107_5"/><i><font color="Green">E84</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:107_5"/><i><font color="Green">E85</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) implies <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E5:107_5"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self">Def13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:107_5"/><i><font color="Green">E87</font></i>: 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:107_5"/><i><font color="Green">E88</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( (  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or  not  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) implies <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E9:107_5"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="comput_1.html#D13" target="_self">Def13</a></i>;<br/><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E15:107"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E15:107"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/><a NAME="E12:107_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107_5"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E8:107_5"><i><font color="Green">E88</font></i></a>, <a class="txt" href="comput_1.html#E9:107_5"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E77"><i><font color="Green">Lemma66</font></i></a></i>;<br/><a NAME="E13:107_5"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107_5"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E78"><i><font color="Green">Lemma67</font></i></a></i>;<br/><a NAME="E14:107_5"/><i><font color="Green">E94</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107_5"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E78"><i><font color="Green">Lemma67</font></i></a></i>;<br/><b>thus </b><a NAME="E15:107_5"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> iff ( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_5_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_5_3_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:107_5_3_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a></i>;<br/><b>assume </b><a NAME="E3:107_5_3_1"/><i><font color="Green">E98</font></i>: 
(  not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_5_3_1_1"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E3:107_5_3_1"><i><font color="Green">E98</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1_1"><b>suppose </b></a><a NAME="E1:107_5_3_1_1_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:107_5_3_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E7:107_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="comput_1.html#E11:107_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E12:107_5"><i><font color="Green">E92</font></i></a></i>;<br/><b>hence </b><a NAME="E3:107_5_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E14:107_5"><i><font color="Green">E94</font></i></a>, <a class="txt" href="comput_1.html#E2:107_5_3_1"><i><font color="Green">E97</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_5_3_1_1_2"><b>suppose </b></a><a NAME="E1:107_5_3_1_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_5_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E13:107_5"><i><font color="Green">E93</font></i></a>, <a class="txt" href="comput_1.html#E14:107_5"><i><font color="Green">E94</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5_3_1"><i><font color="Green">E96</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:107_5_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:107_5_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E15:107"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E7:107_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E11:107_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E12:107_5"><i><font color="Green">E92</font></i></a>, <a class="txt" href="comput_1.html#E13:107_5"><i><font color="Green">E93</font></i></a></i>;<br/>
<a NAME="E4:107_5_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:107_5_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:107_5_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p5"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:107_5_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E16:107_5"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E17:107_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E15:107_5"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E18:107_5"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E15:107"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E5:107_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E7:107_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="comput_1.html#E11:107_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E12:107_5"><i><font color="Green">E92</font></i></a>, <a class="txt" href="comput_1.html#E15:107_5"><i><font color="Green">E95</font></i></a>, <a class="txt" href="comput_1.html#E16:107_5"><i><font color="Green">E96</font></i></a></i>;<br/><a NAME="E19:107_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E20:107_5"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E18:107_5"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:107_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:107_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E10:107_5"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E13:107_5"><i><font color="Green">E93</font></i></a>, <a class="txt" href="comput_1.html#E19:107_5"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E18:107"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> iff ( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_6"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_6_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:107_6_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:107_6_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 ;<br/><a NAME="E4:107_6_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E1:107_6_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E5:107_6_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E5:107"><i><font color="Green">E74</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E6:107_6_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:107_6_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green">Lemma68</font></i></a></i>;<br/><a NAME="E8:107_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E1:107_6_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E6:107_6_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E9:107_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E1:107_6_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:107_6"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:107_6"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E4:107_6"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:107_6"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green">Lemma68</font></i></a></i>;<br/>
<a NAME="E6:107_6"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:107_6"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E4:107_6"><i><font color="Green">E85</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E7:107_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:107_6"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E9:107_6"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E16:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E2:107_6"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E6:107_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:107_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E5:107"><i><font color="Green">E74</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E19:107"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E20:107"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:107"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E21:107"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:107"><i><font color="Green">E71</font></i></a>, <a class="txt" href="comput_1.html#E4:107"><i><font color="Green">E73</font></i></a>, <a class="txt" href="comput_1.html#E79"><i><font color="Green">Lemma68</font></i></a></i>;<br/>
<a NAME="E22:107"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E19:107"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E20:107"><i><font color="Green">E84</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E23:107"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:107"><i><font color="Green">E72</font></i></a>, <a class="txt" href="comput_1.html#E5:107"><i><font color="Green">E74</font></i></a>, <a class="txt" href="comput_1.html#E8:107"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E17:107"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E19:107"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_7.html#T36">FUNCT_7:36</a></i>;<br/>


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