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<span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 16 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default">48 <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"> (  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">D</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <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="b1">C</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#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; <font color="Olive" title="b2">D</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#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; (  for <font color="Olive" title="b3">i</font> being    <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>   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="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 holds <br/>( <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">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">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) &amp; <font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) ) ) ) &amp;  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">D</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <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="b1">C</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#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; <font color="Olive" title="b2">D</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#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; (  for <font color="Olive" title="b3">i</font> being    <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>   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="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 holds <br/>( <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">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">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) &amp; <font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) ) ) ) implies <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> )</span><br/>

<a NAME="E1:84_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 16
 
<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="E2:84_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 16
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:84_1_2"/>
<font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">48 <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 href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 16 </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">s</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">48 <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 href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E5:84_1_2"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 16 )
 ;<br/>
<a NAME="E6:84_1_2"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 16
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_2"><span class="lab"><font color="Green" title="E38">A9</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="kw">given </span><font color="Maroon" title="c5">C1</font>, <font color="Maroon" title="c6">D1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <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><a NAME="E8:84_1_2"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
( <font color="Maroon" title="c5">C1</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#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; <font color="Maroon" title="c6">D1</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#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; (  for <font color="Olive" title="b1">i</font> being    <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>   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> 15 holds <br/>( <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</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> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</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; <font color="Maroon" title="c5">C1</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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">C1</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"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>) &amp; <font color="Maroon" title="c6">D1</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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">D1</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"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></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">C</font>, <font color="Olive" title="b2">D</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <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>  holds <br/>(  not <font color="Olive" title="b1">C</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#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) or  not <font color="Olive" title="b2">D</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#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) or  ex <font color="Olive" title="b3">i</font> being    <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>  st <br/>(  <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="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">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">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) &amp; <font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>) ) ) ) or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">C2</font>, <font color="Maroon" title="c8">D2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <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><a NAME="E10:84_1_2"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
( <font color="Maroon" title="c7">C2</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#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; <font color="Maroon" title="c8">D2</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#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; (  for <font color="Olive" title="b1">i</font> being    <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>   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> 15 holds <br/>( <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</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> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</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; <font color="Maroon" title="c7">C2</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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">C2</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"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>) &amp; <font color="Maroon" title="c8">D2</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#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</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"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></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"> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 implies ( <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <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="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default">$1 <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="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default">$1 <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="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) );<br/>
<a NAME="E11:84_1_2"/><span class="lab"><font color="Green" title="E42">A13</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="84_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E43">A14</font></span>: <font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E44">A15</font></span>: <font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>
;<br/>
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="#E1:84_1_2_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E2:84_1_2_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:84_1_2_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="#E1:84_1_2_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E2:84_1_2_1"><span class="lab"><font color="Green" title="E44">A15</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><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A16</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">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="c9">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:84_1_2_2"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:84_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:84_1_2_2_1"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
(  <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="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">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> 15 )
 ; <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="c2">p</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c3">q</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>

<a NAME="E2:84_1_2_2_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">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#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A20</font></span>: <font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="#E2:84_1_2_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:84_1_2_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">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="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">A21</font></span>: <font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="#E2:84_1_2_2_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E1:84_1_2_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">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="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a></span>
;<br/>
<font color="Maroon" title="c2">p</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E3:84_1_2_2_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E4:84_1_2_2_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E8:84_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">q</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E10:84_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E1:84_1_2_2_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:84_1_2_2_1"/>
( <font color="Maroon" title="c2">p</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c3">q</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="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">C1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c7">C2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">D1</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c8">D2</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E3:84_1_2_2_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E4:84_1_2_2_1"><span class="lab"><font color="Green" title="E48">A21</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E13:84_1_2"/><span class="lab"><font color="Green" title="E44">A22</font></span>: 
 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="#E11:84_1_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E12:84_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>);<br/></span>
<a NAME="E14:84_1_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_1_2_3"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">p</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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">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="txt" href="#E1:84_1_2_3"><span class="lab"><font color="Green" title="E45">A23</font></span></a></span>;<br/>
<a NAME="E3:84_1_2_3"/><span class="lab"><font color="Green" title="E46">A24</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 16 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_2_3"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E2:84_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<a NAME="E5:84_1_2_3"/>
( 1 <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="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c10">i</font> <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> 16 <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="#E3:84_1_2_3"><span class="lab"><font color="Green" title="E46">A24</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="E6:84_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</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="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</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="#E13:84_1_2"><span class="lab"><font color="Green" title="E44">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:84_1_2_3"/>
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:84_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E6:84_1_2"><span class="lab"><font color="Green" title="E39">A10</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>
