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<span class="kw">let </span><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font>, <font color="Maroon" title="c3">j1</font>, <font color="Maroon" title="c4">j2</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="c3">j1</font>,<font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c4">j2</font> <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</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="c3">j1</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,<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c2">i2</font>,<font color="Maroon" title="c4">j2</font> <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a> 
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
( 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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</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="c3">j1</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,<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a> </span><br/>

<a NAME="E3:6"/>
( ( <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; <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j2</font> ) 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> &amp; <font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c4">j2</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="gobrd10.html#D2" target="_self" title="GOBRD10:def.2">Def2</a></span>;<br/>
<a NAME="E4:6"/><span class="kw">then </span>
( ( <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>  &amp; <font color="Maroon" title="c3">j1</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="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 ) or ( <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="c2">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <font color="Maroon" title="c3">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c4">j2</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="#E2:6"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="gobrd10.html#T2" target="_self" title="GOBRD10:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:6"/>
<font color="Maroon" title="c1">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c3">j1</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,<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd10.html#D2" target="_self" title="GOBRD10:def.2">Def2</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/>


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