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<span class="kw">let </span><font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c5">k2</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c2">n</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">k</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c5">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">k</font> ) implies <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:16_1_2"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c5">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">k</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" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font></span><br/>

<a NAME="E3:16_1_2"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E4:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E15">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">j</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" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">k1</font> implies <font color="Maroon" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_1_2_1"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">k1</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" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span><br/><a NAME="E2:16_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E4:16_1_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_2_1"/>
<font color="Maroon" title="c4">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E4:16_1_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E1:16_1_2_1"><span class="lab"><font color="Green" title="E16">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>
<a NAME="E6:16_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16_1_2"/>
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E5:16_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</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/>


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