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<b>let </b><font color="Maroon" title="c1">i0</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c1">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>] <b>means </b>for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Olive" title="b1">i2</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>];<br/>
<a NAME="E2:30_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i><a class="txt" href="int_1.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E3:30_1"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c2">i2</font>
 <b>and </b><br/><a NAME="E2:30_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i2</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c3">i3</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E3:30_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c2">i2</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i3</font>
 ;<br/>

<a NAME="E4:30_1_1"/><b>then </b>
<font color="Maroon" title="c2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:30_1_1"/><b>then </b><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">i3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<b>by </b><i><a class="txt" href="int_1.html#E2:30_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>
<a NAME="E6:30_1_1"/>
<font color="Maroon" title="c3">i3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:30_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="int_1.html#E3:30_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c3">i3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:30_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon" title="c3">i3</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1]
 
<b>by </b><i><a class="txt" href="int_1.html#E2:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="int_1.html#E5:30_1_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:30_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">i3</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:30_1"/><i><font color="Green" title="E23">A10</font></i>: 
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>]
 
<b>from </b><i><a class="ref" href="int_1.html#S2" target="_self" title="INT_1:sch.2">INT_1:sch 2</a>(<a class="txt" href="int_1.html#E2:30_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="int_1.html#E3:30_1"><i><font color="Green" title="E22">A5</font></i></a>);<br/></i>
<a NAME="E5:30_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">i0</font>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:30_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">i0</font>]
 <b>by </b><i><a class="txt" href="int_1.html#E4:30_1"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/>


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