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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span><br/>





<b>set </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c5">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E1:167"/><i><font color="Green" title="E82">F</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#T123" target="_self" title="FUNCT_7:th.123">Thx</a></i>;<br/>
<a NAME="E2:167"/><i><font color="Green" title="E83">H</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_7.html#T120" target="_self" title="FUNCT_7:th.120">Thy</a></i>;<br/>
<b>hence </b><a NAME="E3:167"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:167"><i><font color="Green" title="E82">F</font></i></a>, <a class="ref" href="funct_7.html#T32" target="_self" title="FUNCT_7:th.32">Th32</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E4:167"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E1:167"><i><font color="Green" title="E82">F</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:167_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_1"><b>suppose </b></a><a NAME="E1:167_1_1"/><i><font color="Green" title="E84">S</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="ref" href="funct_7.html#T124" target="_self" title="FUNCT_7:th.124">Th57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:167_1_1"><i><font color="Green" title="E84">S</font></i></a>, <a class="ref" href="funct_7.html#T121" target="_self" title="FUNCT_7:th.121">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:167_1_1"><i><font color="Green" title="E84">S</font></i></a>, <a class="ref" href="funct_7.html#T34" target="_self" title="FUNCT_7:th.34">Th34</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_2"><b>suppose </b></a><a NAME="E1:167_1_2"/><i><font color="Green" title="E84">S</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="ref" href="funct_7.html#T125" target="_self" title="FUNCT_7:th.125">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:167_1_2"><i><font color="Green" title="E84">S</font></i></a>, <a class="txt" href="#E2:167"><i><font color="Green" title="E83">H</font></i></a>, <a class="ref" href="funct_7.html#T33" target="_self" title="FUNCT_7:th.33">Th33</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="167_1_3"><b>suppose </b></a><a NAME="E1:167_1_3"/><i><font color="Green" title="E84">S</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="ref" href="funct_7.html#T126" target="_self" title="FUNCT_7:th.126">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:167_1_3"><i><font color="Green" title="E84">S</font></i></a>, <a class="ref" href="funct_7.html#T122" target="_self" title="FUNCT_7:th.122">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_7.html#K11" title="FUNCT_7:func.11">followed_by</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> 1,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="#E1:167_1_3"><i><font color="Green" title="E84">S</font></i></a>, <a class="ref" href="funct_7.html#T34" target="_self" title="FUNCT_7:th.34">Th34</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
