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<span class="kw">let </span><font color="Maroon" title="c1">a</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#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">Alg</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p2">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b2">h</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> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">Alg</font>) holds <br/> <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <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"><font color="Olive" title="b2">h</font> <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> <font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Alg</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p2">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</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> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">Alg</font>) holds <br/> <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <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"><font color="Olive" title="b1">h</font> <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> <font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c3">h</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 href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</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="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">Alg</font>) implies  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <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"><font color="Maroon" title="c3">h</font> <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> <font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )</span><br/>





<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E25">HC00</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K6" title="BINPACK1:func.6">OnlinePackingHistory</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">Alg</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">  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <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"><font color="Maroon" title="c3">h</font> <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> <font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/>

<a NAME="E2:31"/><span class="lab"><font color="Green" title="E26">L010</font></span>: 
<font color="Maroon" title="c3">h</font> <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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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="#E1:31"><span class="lab"><font color="Green" title="E25">HC00</font></span></a>, <a class="ref" href="binpack1.html#D5" target="_self" onmouseover="rs('binpack1/D5')" onmouseout="rh()">defPackHistory</a></span>;<br/>
<a NAME="E3:31"/>
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> 1
 
;<br/>
<a NAME="E4:31"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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> <font color="Maroon" title="c1">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="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E5:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">L049</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">L055</font></span>:  <a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) = 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>

<br/>.= 
<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"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E27">L049</font></span></a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_3.html#T157" onmouseover="rs('finseq_3/T157')" onmouseout="rh()">FINSEQ_3:157</a></span>
;<br/>
<span class="kw">thus </span> <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <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"><font color="Maroon" title="c3">h</font> <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="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</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>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E26">L010</font></span></a></span>
<br/>.= 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binpack1.html#T3" target="_self" onmouseover="rs('binpack1/T3')" onmouseout="rh()">NF501</a></span>
<br/>.= 
<font color="Maroon" title="c1">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="txt" href="#E6:31"><span class="lab"><font color="Green" title="E28">L055</font></span></a>, <a class="ref" href="rvsum_1.html#T73" onmouseover="rs('rvsum_1/T73')" onmouseout="rh()">RVSUM_1:73</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/>


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