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

<span class="kw">let </span><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> implies <font color="Maroon" title="c1">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c2">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> 
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <span class="kw">and </span><br/><a NAME="E3:3"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c2">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> </span><br/>

<a NAME="E4:3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E5:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c1">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E6:3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E7:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c2">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E8:3"/>
( <font color="Maroon" title="c2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="gobrd10.html#D1" target="_self" title="GOBRD10:def.1">Def1</a></span>;<br/>
<a NAME="E9:3"/><span class="kw">then </span>
( <font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 or <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
;<br/>
<span class="kw">hence </span><a NAME="E10:3"/>
<font color="Maroon" title="c1">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c2">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E7:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="gobrd10.html#D1" target="_self" title="GOBRD10:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
