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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span><br/>



<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><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E1:38"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 for <font color="Olive" title="b1">x</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">x</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="lab"><font color="Green" title="E22">A2</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" title="NAT_D:th.67">NAT_D:67</a></span>
;<br/>
<span class="lab"><font color="Green" title="E23">A3</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" title="NAT_D:th.67">NAT_D:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T65" title="NAT_D:th.65">NAT_D:65</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E3:38_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">hence </span><a NAME="E4:38_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</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="#E1:38_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:38"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/>
<a NAME="E3:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">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="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/>
<a NAME="E4:38"/>
 for <font color="Olive" title="b1">n</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">n</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="#E3:38"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E21">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:38"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
