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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b1">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>);<br/>
<a NAME="E1:33"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T11" onmouseover="rs('int_1/T11')" onmouseout="rh()">INT_1:11</a></span>;<br/>
<a NAME="E2:33"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<a NAME="E1:33_1"/>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_6.html#T18" target="_self" onmouseover="rs('hilb10_6/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:33_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T15" onmouseover="rs('int_1/T15')" onmouseout="rh()">INT_1:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:33"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E34">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:33"/>
 <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>), <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
