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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/> <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">g</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">i</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span> implies  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> )</span><br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">fg</font> = <font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:10"/>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c4">fg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E4:10"/><b>then </b>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c4">fg</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c4">fg</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E6:10"/>
 <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
 <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">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="c2">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:10"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c4">fg</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></i>;<br/>
<a NAME="E9:10"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<a NAME="E10:10"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
 <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c4">fg</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E5:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E8:10"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<a NAME="E11:10"/><b>then </b>
<font color="Maroon" title="c4">fg</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c4">fg</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c4">fg</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:10"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c4">fg</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E5:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<a NAME="E13:10"/><b>then </b>
 <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c4">fg</font>,<font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E14:10"/>
 <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="#E10:10"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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