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<a NAME="E1:30_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c1">i0</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="c1">i0</font>]</span><br/>

<div><i><font color="Green" title="E21">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
<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">i0</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="c1">i0</font>]</span><br/><a NAME="E2:30_1_1"/><i><font color="Green" title="E23">A6</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 <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">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E3:30_1_1"/>
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <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">P<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="#E1:30_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E2:30_1_1"><i><font color="Green" title="E23">A6</font></i></a>);<br/></i><b>hence </b><a NAME="E4:30_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">i0</font>]
 <b>by </b><i><a class="txt" href="#E1:30_1_1"><i><font color="Green" title="E22">A5</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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green" title="E22">A7</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>()
 ; <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="c1">i0</font>]</span><br/><a NAME="E2:30_1_2"/><i><font color="Green" title="E23">A8</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 <font color="Olive" title="b1">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1]
 <b>by </b><i><a class="txt" href="#E2:30"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E3:30_1_2"/>
for <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>]
 <b>from </b><i><a class="ref" href="int_1.html#S3" target="_self" title="INT_1:sch.3">INT_1:sch 3</a>(<a class="txt" href="#E1:30_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E2:30_1_2"><i><font color="Green" title="E23">A8</font></i></a>);<br/></i><b>hence </b><a NAME="E4:30_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">i0</font>]
 <b>by </b><i><a class="txt" href="#E1:30_1_2"><i><font color="Green" title="E22">A7</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">i0</font>]
 <b>by </b><i><a class="txt" href="#E2:30_1"><i><font color="Green" title="E21">A4</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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