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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1];<br/>
<b>consider </b><font color="Maroon" title="c1">t</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">t</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">s</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">s</font> = <font color="Maroon" title="c2">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:1_1"/>
<font color="Maroon" title="c1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="comptrig.html#E4:1_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<a NAME="E7:1_1"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="comptrig.html#E2:1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/>
<div><i><font color="Green" title="E6">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1"/><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:1_1_1"/><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E4:1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="comptrig.html#E1:1_1_1"><i><font color="Green" title="E7">A7</font></i></a></i>;<br/><a NAME="E5:1_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">n</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:1_1_1"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k1</font>
 <b>and </b><br/><a NAME="E8:1_1_1"/><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="comptrig.html#E2:1_1_1"><i><font color="Green" title="E8">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_1"/><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">l</font> = <font color="Maroon" title="c7">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>take </b><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c8">l</font>;<br/><b>thus </b><a NAME="E12:1_1_1"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="comptrig.html#E7:1_1_1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="comptrig.html#E10:1_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E13:1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">l</font>]
 <b>by </b><i><a class="txt" href="comptrig.html#E8:1_1_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="comptrig.html#E10:1_1_1"><i><font color="Green" title="E11">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="comptrig.html#E7:1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="comptrig.html#E8:1_1"><i><font color="Green" title="E6">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:1_1"/>
<font color="Maroon">P<sub>1</sub></font>[1]
 ;<br/>


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