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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st ( ( <font color="Olive" title="b1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Olive" title="b1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) &amp; <font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">h</font>, <font color="Maroon" title="c3">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c2">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) &amp; <font color="Maroon" title="c2">h</font>,<font color="Maroon" title="c3">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> implies  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">i</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( ( <font color="Maroon" title="c2">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c2">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">h</font>,<font color="Maroon" title="c3">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">i</font></span><br/>

<a NAME="E3:35"/>
<font color="Maroon" title="c2">h</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:35"/>
 <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="binari_4.html#E34"><span class="lab"><font color="Green" title="E22">Lm5</font></span></a>, <a class="txt" href="binari_4.html#E35"><span class="lab"><font color="Green" title="E23">Lm6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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