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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">RK</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">k</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b3">IP</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b4">M</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> holds   <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Olive" title="b4">M</font>,<font color="Olive" title="b2">F</font>,<font color="Olive" title="b3">IP</font>,<font color="Olive" title="b1">RK</font>)</span>)</span>,<font color="Olive" title="b2">F</font>,<font color="Olive" title="b3">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Olive" title="b1">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">RK</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">k</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</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">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b2">IP</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b3">M</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> holds   <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</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">IP</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><br/>  for <font color="Olive" title="b2">M</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> holds   <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Olive" title="b2">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">IP</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</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">M</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> holds   <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">M</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</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="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font></span><br/>









<span class="kw">set </span><font color="Maroon" title="c8">REVRKS</font> =  <a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">EM</font> =  <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">L</font>, <font color="Maroon" title="c11">R</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
( <font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K4" title="DESCIP_1:func.4">SP-Left</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">M</font></span>)</span> &amp; <font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K5" title="DESCIP_1:func.5">SP-Right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">M</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/>( <font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">RK</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp;  <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D29" target="_self" onmouseover="rs('descip_1/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">LB</font>, <font color="Maroon" title="c13">RB</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:87"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K4" title="DESCIP_1:func.4">SP-Left</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K5" title="DESCIP_1:func.5">SP-Right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/>( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp;  <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D29" target="_self" onmouseover="rs('descip_1/D29')" onmouseout="rh()">Def29</a></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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies ( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1</span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1</span>)</span> ) );<br/>
<a NAME="E5:87"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:87_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E2:87_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">IP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E3:87_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">IP</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" onmouseover="rs('finseq_2/T131')" onmouseout="rh()">FINSEQ_2:131</a></span>;<br/>
<a NAME="E4:87_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>;<br/>
<a NAME="E5:87_1"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T21" onmouseover="rs('finseq_1/T21')" onmouseout="rh()">FINSEQ_1:21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:87_1"/>
<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="txt" href="#E4:87"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="finseq_5.html#T37" onmouseover="rs('finseq_5/T37')" onmouseout="rh()">FINSEQ_5:37</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="E6:87"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
 for <font color="Olive" title="b1">i</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">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</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="c14">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">i</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="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_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"> ( <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies ( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:87_2_1"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</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="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><a NAME="E2:87_2_1"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">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="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:87_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:87_2_1"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E5:87_2_1"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E6:87_2_1"/>
<font color="Maroon" title="c14">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_1"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">i16</font> = <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">i161</font> = <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_2_1"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E9:87_2_1"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E10:87_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A16</font></span>: 
( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">Ki</font> = <span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/><span class="lab"><font color="Green" title="E52">A17</font></span>: <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c14">i</font> = 
<font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E2:87_2_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c16">i161</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:87_2_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a></span>
;<br/><a NAME="E13:87_2_1"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">RK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/><a NAME="E14:87_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E15:87_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 ;<br/><a NAME="E16:87_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A19</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">RK</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_2_1"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E17:87_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T57" onmouseover="rs('finseq_5/T57')" onmouseout="rh()">FINSEQ_5:57</a></span>;<br/><a NAME="E18:87_2_1"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E19:87_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_2_1"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E20:87_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">i16</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_2_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a></span>;<br/><a NAME="E21:87_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">RK</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_2_1"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="lab"><font color="Green" title="E58">A23</font></span>: <span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:87_2_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">RK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">RK</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:87_2_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="finseq_5.html#T58" onmouseover="rs('finseq_5/T58')" onmouseout="rh()">FINSEQ_5:58</a></span>
<br/>.= 
<font color="Maroon" title="c4">RK</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:87_2_1"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E13:87_2_1"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/><font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c14">i</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c16">i161</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c16">i161</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">RK</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i161</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:87_2_1"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E1:87_2_1"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E2:87_2_1"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c16">i161</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c16">i161</font></span>)</span>,<font color="Maroon" title="c17">Ki</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:87_2_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E24:87_2_1"/>
( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2_1"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="txt" href="#E12:87_2_1"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="descip_1.html#T17" target="_self" onmouseover="rs('descip_1/T17')" onmouseout="rh()">Th17</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>
<span class="kw">hence </span><a NAME="E3:87_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E7:87"/>
 for <font color="Olive" title="b1">i</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">i</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="#E5:87"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E43">A8</font></span></a>);<br/></span>
<a NAME="E8:87"/><span class="kw">then </span>
( <font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 
;<br/>
<a NAME="E9:87"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">RB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">LB</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#T3" target="_self" onmouseover="rs('descip_1/T3')" onmouseout="rh()">Th3</a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:87"/>
 <a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K39" title="DESCIP_1:func.39">DES-like-CoDec</a> (<font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span>,<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">IP</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K1" title="DESCIP_1:func.1">Rev</a> <font color="Maroon" title="c4">RK</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</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>
