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<span class="kw">let </span><font color="Maroon" title="c1">n</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">  for <font color="Olive" title="b1">epsilon</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">a</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive</a>   <a href="binpack1.html#V1" title="BINPACK1:attr.1">at_most_one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b3">f</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">epsilon</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b4">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies <font color="Olive" title="b2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">epsilon</font> ) &amp; ( <font color="Olive" title="b4">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <font color="Olive" title="b2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">epsilon</font> ) ) ) &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K7" title="BINPACK1:func.7">OnlinePacking</a> (<font color="Olive" title="b2">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Olive" title="b2">a</font></span>)</span>) holds <br/><font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">epsilon</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">a</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive</a>   <a href="binpack1.html#V1" title="BINPACK1:attr.1">at_most_one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b2">f</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">epsilon</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b3">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; ( <font color="Olive" title="b3">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) ) ) &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K7" title="BINPACK1:func.7">OnlinePacking</a> (<font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Olive" title="b1">a</font></span>)</span>) holds <br/><font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive</a>   <a href="binpack1.html#V1" title="BINPACK1:attr.1">at_most_one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">f</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">epsilon</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b2">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; ( <font color="Olive" title="b2">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) ) ) &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K7" title="BINPACK1:func.7">OnlinePacking</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>) holds <br/><font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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"> ( <font color="Maroon" title="c1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">epsilon</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b1">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies <font color="Maroon" title="c3">a</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> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; ( <font color="Olive" title="b1">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <font color="Maroon" title="c3">a</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> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) ) ) &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K7" title="BINPACK1:func.7">OnlinePacking</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>) implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56"/><span class="lab"><font color="Green" title="E47">L010</font></span>: 
<font color="Maroon" title="c1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">and </span><br/><a NAME="E2:56"/><span class="lab"><font color="Green" title="E48">L030</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:56"/><span class="lab"><font color="Green" title="E49">L040</font></span>: 
<font color="Maroon" title="c2">epsilon</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:56"/><span class="lab"><font color="Green" title="E50">L050</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( <font color="Olive" title="b1">i</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies <font color="Maroon" title="c3">a</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> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; ( <font color="Olive" title="b1">i</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <font color="Maroon" title="c3">a</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> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) )
 <span class="kw">and </span><br/><a NAME="E5:56"/><span class="lab"><font color="Green" title="E51">HL00</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K7" title="BINPACK1:func.7">OnlinePacking</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</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="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<a NAME="E6:56"/><span class="lab"><font color="Green" title="E52">L020</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E47">L010</font></span></a>, <a class="ref" href="binpack1.html#T1" target="_self" onmouseover="rs('binpack1/T1')" onmouseout="rh()">NF992</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">h</font> =  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></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>] <span class="kw">means </span>( ( $1 is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; ( $1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1 );<br/>
<a NAME="E7:56"/><span class="lab"><font color="Green" title="E53">L300</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E54">L305</font></span>: 
<span class="p1">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binpack1.html#D5" target="_self" onmouseover="rs('binpack1/D5')" onmouseout="rh()">defPackHistory</a></span>;<br/>
<a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E55">L324</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E3:56_1"/><span class="lab"><font color="Green" title="E56">L332</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E52">L020</font></span></a></span>;<br/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) = 
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="binpack1.html#T20" target="_self" onmouseover="rs('binpack1/T20')" onmouseout="rh()">NF502</a></span>
<br/>.= 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E55">L324</font></span></a>, <a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E56">L332</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E50">L050</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:56_1"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E54">L305</font></span></a>, <a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E55">L324</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>
<a NAME="E8:56"/><span class="lab"><font color="Green" title="E54">L400</font></span>: 
 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 1 <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> &amp; <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i0</font> be    <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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i0</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_2"/><span class="lab"><font color="Green" title="E55">L410</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">and </span><br/><a NAME="E2:56_2"/><span class="lab"><font color="Green" title="E56">L411</font></span>: 
<font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E3:56_2"/><span class="lab"><font color="Green" title="E57">L412</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i0</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="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">i0p</font> = <font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E5:56_2"/><span class="lab"><font color="Green" title="E58">L469</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E6:56_2"/><span class="lab"><font color="Green" title="E59">L480</font></span>: 
<font color="Maroon" title="c7">i0p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E7:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">L490</font></span>: 
<font color="Maroon" title="c7">i0p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2"><span class="lab"><font color="Green" title="E58">L469</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">hi0</font> = <span class="p1">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T21" target="_self" onmouseover="rs('binpack1/T21')" onmouseout="rh()">NF505</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">hi0p</font> = <span class="p1">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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="#E5:56_2"><span class="lab"><font color="Green" title="E58">L469</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a>, <a class="txt" href="#E6:56_2"><span class="lab"><font color="Green" title="E59">L480</font></span></a>, <a class="ref" href="binpack1.html#T21" target="_self" onmouseover="rs('binpack1/T21')" onmouseout="rh()">NF505</a></span>;<br/>
<a NAME="E10:56_2"/><span class="lab"><font color="Green" title="E61">L496</font></span>: 
( <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">hi0</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<font color="Maroon" title="c8">hi0</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<font color="Maroon" title="c8">hi0</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a></span>;<br/>
<a NAME="E11:56_2"/><span class="lab"><font color="Green" title="E62">L498</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">hi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T22" target="_self" onmouseover="rs('binpack1/T22')" onmouseout="rh()">NF510</a></span>;<br/>
<a NAME="E12:56_2"/><span class="lab"><font color="Green" title="E63">L500</font></span>: 
( <font color="Maroon" title="c7">i0p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies ( <font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_1"/><span class="lab"><font color="Green" title="E64">L510</font></span>: 
<font color="Maroon" title="c7">i0p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> )</span><br/>

<a NAME="E2:56_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E50">L050</font></span></a>, <a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E60">L490</font></span></a></span>;<br/>
<a NAME="E3:56_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">L542</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">epsilon</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_1"><span class="lab"><font color="Green" title="E64">L510</font></span></a>, <a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a></span>;<br/>
<a NAME="E4:56_2_1"/><span class="lab"><font color="Green" title="E66">L549</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">epsilon</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E49">L040</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="lab"><font color="Green" title="E67">L580</font></span>: <font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> = 
<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<font color="Maroon" title="c8">hi0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_1"><span class="lab"><font color="Green" title="E66">L549</font></span></a>, <a class="txt" href="#E3:56_2_1"><span class="lab"><font color="Green" title="E65">L542</font></span></a>, <a class="txt" href="#E11:56_2"><span class="lab"><font color="Green" title="E62">L498</font></span></a>, <a class="ref" href="binpack1.html#D7" target="_self" onmouseover="rs('binpack1/D7')" onmouseout="rh()">defNextFit</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:56_2_1"/>
<font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</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"> (  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font> )</span><br/>

<span class="kw">thus </span> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c8">hi0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T24" target="_self" onmouseover="rs('binpack1/T24')" onmouseout="rh()">NF525</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_1"><span class="lab"><font color="Green" title="E67">L580</font></span></a>, <a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a>, <a class="ref" href="finseq_1.html#T9" onmouseover="rs('finseq_1/T9')" onmouseout="rh()">FINSEQ_1:9</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">  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font></span><br/>

<a NAME="E8:56_2_1"/>
 not <font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:56_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">L590</font></span>: 
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a>, <a class="ref" href="binpack1.html#T13" target="_self" onmouseover="rs('binpack1/T13')" onmouseout="rh()">NF290</a></span>;<br/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) = 
<span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">hi0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_2"><span class="lab"><font color="Green" title="E61">L496</font></span></a>, <a class="txt" href="#E11:56_2"><span class="lab"><font color="Green" title="E62">L498</font></span></a>, <a class="txt" href="#E6:56_2"><span class="lab"><font color="Green" title="E59">L480</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a>, <a class="ref" href="binpack1.html#T8" target="_self" onmouseover="rs('binpack1/T8')" onmouseout="rh()">NF200</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_1"><span class="lab"><font color="Green" title="E67">L580</font></span></a>, <a class="txt" href="#E9:56_2_1"><span class="lab"><font color="Green" title="E68">L590</font></span></a>, <a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T22" target="_self" onmouseover="rs('binpack1/T22')" onmouseout="rh()">NF510</a></span>
<br/>.= 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E60">L490</font></span></a>, <a class="txt" href="#E1:56_2_1"><span class="lab"><font color="Green" title="E64">L510</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E50">L050</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:56_2_1"/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p3">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p5">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">epsilon</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>
<a NAME="E13:56_2"/>
( <font color="Maroon" title="c7">i0p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies ( <font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_2"/><span class="lab"><font color="Green" title="E64">L710</font></span>: 
<font color="Maroon" title="c7">i0p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> )</span><br/>

<a NAME="E2:56_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">L720</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E50">L050</font></span></a>, <a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E60">L490</font></span></a></span>;<br/>
<a NAME="E3:56_2_2"/><span class="lab"><font color="Green" title="E66">L742</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">epsilon</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2"><span class="lab"><font color="Green" title="E65">L720</font></span></a>, <a class="txt" href="#E1:56_2_2"><span class="lab"><font color="Green" title="E64">L710</font></span></a>, <a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a></span>;<br/>
<a NAME="E4:56_2_2"/><span class="lab"><font color="Green" title="E67">L749</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">epsilon</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E49">L040</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">L780</font></span>: <font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> = 
<span class="p1">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span>,<font color="Maroon" title="c8">hi0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T23" target="_self" onmouseover="rs('binpack1/T23')" onmouseout="rh()">NF520</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2"><span class="lab"><font color="Green" title="E66">L742</font></span></a>, <a class="txt" href="#E11:56_2"><span class="lab"><font color="Green" title="E62">L498</font></span></a>, <a class="ref" href="binpack1.html#D7" target="_self" onmouseover="rs('binpack1/D7')" onmouseout="rh()">defNextFit</a>, <a class="txt" href="#E4:56_2_2"><span class="lab"><font color="Green" title="E67">L749</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:56_2_2"/>
<font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i0p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</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"> (  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font> &amp;  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font> )</span><br/>

<span class="kw">thus </span> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">hi0p</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c8">hi0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T24" target="_self" onmouseover="rs('binpack1/T24')" onmouseout="rh()">NF525</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">i0p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_2"><span class="lab"><font color="Green" title="E68">L780</font></span></a>, <a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a>, <a class="ref" href="finseq_1.html#T9" onmouseover="rs('finseq_1/T9')" onmouseout="rh()">FINSEQ_1:9</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">  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font></span><br/>

<a NAME="E8:56_2_2"/>
 not <font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:56_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">L790</font></span>: 
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">hi0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E57">L412</font></span></a>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a>, <a class="ref" href="binpack1.html#T13" target="_self" onmouseover="rs('binpack1/T13')" onmouseout="rh()">NF290</a></span>;<br/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) = 
<span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">hi0</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">hi0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_2"><span class="lab"><font color="Green" title="E61">L496</font></span></a>, <a class="txt" href="#E11:56_2"><span class="lab"><font color="Green" title="E62">L498</font></span></a>, <a class="txt" href="#E6:56_2"><span class="lab"><font color="Green" title="E59">L480</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a>, <a class="ref" href="binpack1.html#T8" target="_self" onmouseover="rs('binpack1/T8')" onmouseout="rh()">NF200</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_2"><span class="lab"><font color="Green" title="E68">L780</font></span></a>, <a class="txt" href="#E9:56_2_2"><span class="lab"><font color="Green" title="E69">L790</font></span></a>, <a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E55">L410</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E56">L411</font></span></a>, <a class="ref" href="binpack1.html#T22" target="_self" onmouseover="rs('binpack1/T22')" onmouseout="rh()">NF510</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E60">L490</font></span></a>, <a class="txt" href="#E1:56_2_2"><span class="lab"><font color="Green" title="E64">L710</font></span></a>, <a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E50">L050</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:56_2_2"/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">hi0p</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">hi0p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">epsilon</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="E14:56_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E63">L500</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>
<a NAME="E9:56"/><span class="lab"><font color="Green" title="E55">L899</font></span>: 
 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 1 <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> holds <br/><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="int_1.html#S7" onmouseover="rs('int_1/S7')" onmouseout="rh()">INT_1:sch 7</a>(<a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E53">L300</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E54">L400</font></span></a>);<br/></span>
<a NAME="E10:56"/><span class="lab"><font color="Green" title="E56">L902</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="binpack1.html#K8" title="BINPACK1:func.8">NextFit</a> <font color="Maroon" title="c3">a</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E51">HL00</font></span></a>, <a class="ref" href="binpack1.html#D5" target="_self" onmouseover="rs('binpack1/D5')" onmouseout="rh()">defPackHistory</a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">n</font> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" onmouseover="rs('finseq_1/T57')" onmouseout="rh()">FINSEQ_1:57</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E52">L020</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E48">L030</font></span></a>, <a class="txt" href="#E9:56"><span class="lab"><font color="Green" title="E55">L899</font></span></a>, <a class="txt" href="#E10:56"><span class="lab"><font color="Green" title="E56">L902</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>
