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<b>consider </b><font color="Maroon" title="c1">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E2:32_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">i1</font>]
 <b>by </b><i><a class="txt" href="int_1.html#E2:32"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E3:32_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:32"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_1.html#E2:32_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E21">A4</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">F<sub>1</sub></font>() <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E31"><i><font color="Green" title="E16">Lm6</font></i></a></i>;<br/>
<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>[<font color="Maroon">F<sub>1</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1];<br/>
<a NAME="E5:32_1"/><i><font color="Green" title="E22">A5</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="int_1.html#E2:32_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="int_1.html#E4:32_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">l</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c2">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="int_1.html#E5:32_1"><i><font color="Green" title="E22">A5</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c3">i0</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font>] &amp; ( for <font color="Olive" title="b1">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i1</font>] holds <br/><font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i1</font> ) )</span><br/>

<a NAME="E8:32_1"/>
for <font color="Olive" title="b1">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i1</font>] holds <br/><font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i1</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">i1</font>] implies <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font> )</span><br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">i1</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font></span><br/>

<a NAME="E2:32_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:32"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">n</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> <b> such that </b><br/><a NAME="E4:32_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i1</font>
 <b>by </b><i><a class="txt" href="int_1.html#E31"><i><font color="Green" title="E16">Lm6</font></i></a></i>;<br/>
<a NAME="E5:32_1_1"/>
<font color="Maroon" title="c3">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="int_1.html#E7:32_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="int_1.html#E1:32_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="int_1.html#E4:32_1_1"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_1_1"/>
<font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:32_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">i0</font>] &amp; ( for <font color="Olive" title="b1">i1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i1</font>] holds <br/><font color="Maroon" title="c3">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i1</font> ) )
 <b>by </b><i><a class="txt" href="int_1.html#E7:32_1"><i><font color="Green" title="E23">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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