<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</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">S</font> being   <a href="field_4.html#NM1" title="FIELD_4:NM.1">RingExtension</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/> for <font color="Olive" title="b4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b4">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Olive" title="b3">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="field_4.html#NM1" title="FIELD_4:NM.1">RingExtension</a> of <font color="Maroon" title="c1">R</font>; <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   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> for <font color="Olive" title="b3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b3">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <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">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/> for <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b2">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Olive" title="b1">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <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" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> implies  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b1">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E8">AS</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ; <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">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b1">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">i</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"> <font color="Maroon" title="c5">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E2:18"/><span class="lab"><font color="Green" title="E9">K</font></span>: 
<font color="Maroon" title="c1">R</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>] <span class="kw">means </span> for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</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"> ( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T59" onmouseover="rs('ring_3/T59')" onmouseout="rh()">RING_3:59</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E9">K</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c6">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="ring_3.html#T59" onmouseover="rs('ring_3/T59')" onmouseout="rh()">RING_3:59</a></span>
;<br/> <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="E4:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<span class="lab"><font color="Green" title="E11">A6</font></span>: 1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E8">AS</font></span></a>, <a class="ref" href="ring_3.html#T60" onmouseover="rs('ring_3/T60')" onmouseout="rh()">RING_3:60</a></span>
<br/>.= 
1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T60" onmouseover="rs('ring_3/T60')" onmouseout="rh()">RING_3:60</a></span>
;<br/>
<a NAME="E6:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A7</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E9">K</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/>
<a NAME="E7:18"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 for <font color="Olive" title="b1">u</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</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="18_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">u</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font>]
 ; <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="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<a NAME="E2:18_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_3_1"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</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"> ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_3_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <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" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T64" onmouseover="rs('ring_3/T64')" onmouseout="rh()">RING_3:64</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E9">K</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>
<br/>.= 
<font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="ring_3.html#T64" onmouseover="rs('ring_3/T64')" onmouseout="rh()">RING_3:64</a></span>
;<br/> <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>
<span class="kw">hence </span><a NAME="E4:18_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1]
 ; <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="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_3_2"><span class="kw">now </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">  for <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/><font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">k</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"> ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_3_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <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" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T62" onmouseover="rs('ring_3/T62')" onmouseout="rh()">RING_3:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E9">K</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>
<br/>.= 
<font color="Maroon" title="c7">k</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="ring_3.html#T62" onmouseover="rs('ring_3/T62')" onmouseout="rh()">RING_3:62</a></span>
;<br/> <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>
<span class="kw">hence </span><a NAME="E6:18_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ; <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="E8:18"/>
 for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="int_1.html#S4" onmouseover="rs('int_1/S4')" onmouseout="rh()">INT_1:sch 4</a>(<a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:18"/>
<font color="Maroon" title="c5">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c4">b</font>
 ; <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>
