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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> $1];<br/>
<a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E25">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:38_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><span class="kw">hence </span><a NAME="E3:38_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]</span><br/><div class="add"><a NAME="E2:38_1_2_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_2"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]</span><br/><div class="add"><a NAME="E2:38_1_2_2"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></span>;<br/><a NAME="E3:38_1_2_2"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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">F<sub>1</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:38_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E1:38_1_2_2"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E2:38_1_2_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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