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<b>let </b><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> ;<br/>







<b>assume </b><a NAME="E1:5"/>
<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> 
 ;<br/>

<a NAME="E2:5"/><b>then </b>
( ( <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>  ) )
 
<b>by </b><i><a class="ref" href="gobrd10.html#D2" target="_self" title="GOBRD10:def.2">Def2</a></i>;<br/>
<a NAME="E3:5"/><b>then </b>
( ( <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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 &amp; <font color="Maroon" title="c3">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  ) )
 
<b>by </b><i><a class="ref" href="gobrd10.html#T1" target="_self" title="GOBRD10:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c3">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a> 
 <b>by </b><i><a class="ref" href="gobrd10.html#D2" target="_self" title="GOBRD10:def.2">Def2</a></i>;<br/>


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