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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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="c1">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="c2">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="c1">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="c2">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="c2">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="c2">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>  ex <font color="Olive" title="b5">OUT</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="c2">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> ex <font color="Olive" title="b6">L</font>, <font color="Olive" title="b7">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="c2">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> st <br/>( <font color="Olive" title="b6">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="Olive" title="b3">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">M</font></span>)</span> &amp; <font color="Olive" title="b7">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="Olive" title="b3">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">M</font></span>)</span> &amp; (  for <font color="Olive" title="b8">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="b8">i</font> &amp; <font color="Olive" title="b8">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="Olive" title="b6">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">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="Olive" title="b7">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b8">i</font> &amp; <font color="Olive" title="b7">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">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="Olive" title="b6">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b7">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b8">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">RK</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b5">OUT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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="Olive" title="b7">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="Olive" title="b6">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><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="c1">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="c2">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="c1">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="c2">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="c2">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="c2">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>  ex <font color="Olive" title="b4">OUT</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="c2">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> ex <font color="Olive" title="b5">L</font>, <font color="Olive" title="b6">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="c2">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> st <br/>( <font color="Olive" title="b5">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="Olive" title="b2">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">M</font></span>)</span> &amp; <font color="Olive" title="b6">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="Olive" title="b2">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">M</font></span>)</span> &amp; (  for <font color="Olive" title="b7">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="b7">i</font> &amp; <font color="Olive" title="b7">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="Olive" title="b5">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">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="Olive" title="b6">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b6">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">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="Olive" title="b5">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b6">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b7">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="b7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b4">OUT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b6">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="Olive" title="b5">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><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="c2">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="c1">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="c2">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="c2">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="c2">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>  ex <font color="Olive" title="b3">OUT</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="c2">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> ex <font color="Olive" title="b4">L</font>, <font color="Olive" title="b5">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="c2">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> st <br/>( <font color="Olive" title="b4">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="Olive" title="b1">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">M</font></span>)</span> &amp; <font color="Olive" title="b5">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="Olive" title="b1">IP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">M</font></span>)</span> &amp; (  for <font color="Olive" title="b6">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="b6">i</font> &amp; <font color="Olive" title="b6">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="Olive" title="b4">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">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="Olive" title="b5">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b5">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">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="Olive" title="b4">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b6">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="Olive" title="b5">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b6">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="b6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b3">OUT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b5">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="Olive" title="b4">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><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="c2">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="c2">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>  ex <font color="Olive" title="b2">OUT</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="c2">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> ex <font color="Olive" title="b3">L</font>, <font color="Olive" title="b4">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="c2">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> st <br/>( <font color="Olive" title="b3">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="Olive" title="b1">M</font></span>)</span> &amp; <font color="Olive" title="b4">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="Olive" title="b1">M</font></span>)</span> &amp; (  for <font color="Olive" title="b5">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="b5">i</font> &amp; <font color="Olive" title="b5">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="Olive" title="b3">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="Olive" title="b4">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="Olive" title="b3">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b5">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="Olive" title="b4">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b5">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="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b2">OUT</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="Olive" title="b4">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="Olive" title="b3">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><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="c2">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">  ex <font color="Olive" title="b1">OUT</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="c2">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> ex <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">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="c2">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> st <br/>( <font color="Olive" title="b2">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="Olive" title="b3">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="b4">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="b4">i</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="Olive" title="b2">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">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="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">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="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b1">OUT</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="Olive" title="b3">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="Olive" title="b2">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><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>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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 class="kw">means </span>( $4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> ($2,<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> ($3,<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">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) );<br/>
<a NAME="E1:85"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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>  ex <font color="Olive" title="b4">x1</font>, <font color="Olive" title="b5">y1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">x1</font>,<font color="Olive" title="b5">y1</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">L</font>, <font color="Maroon" title="c9">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="c2">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="E3:85"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
( <font color="Maroon" title="c8">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="c9">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">n</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">n</font>,<font color="Maroon" title="c8">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c9">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">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">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c9">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">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_2.html#S3" onmouseover="rs('recdef_2/S3')" onmouseout="rh()">RECDEF_2:sch 3</a>(<a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E38">A1</font></span></a>);<br/></span>
<a NAME="E4:85"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 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="c8">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="c9">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="c9">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="c8">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="c9">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>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E39">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">RL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c8">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c3">k</font></span>)</span> as    <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="c2">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="finseq_2.html#T131" onmouseover="rs('finseq_2/T131')" onmouseout="rh()">FINSEQ_2:131</a></span>;<br/>
<a NAME="E6:85"/>
<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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">RL</font> is    <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="c2">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>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:85"/>
 ex <font color="Olive" title="b1">OUT</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="c2">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> ex <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">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="c2">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> st <br/>( <font color="Olive" title="b2">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="Olive" title="b3">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="b4">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="b4">i</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="Olive" title="b2">L</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">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="Olive" title="b3">R</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">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="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span>) ) ) &amp; <font color="Olive" title="b1">OUT</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="Olive" title="b3">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="Olive" title="b2">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="txt" href="#E4:85"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E39">A2</font></span></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>
