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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:89_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:89_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_1_1"/>
not <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_1_1_1"/>
( <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:89_1_1_1_1_1"/>
<font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:89_1_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:89_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:89_1_1_1_1_2"/>
not <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:89_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:89_1"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" target="_self">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:89_1"/>
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/>


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