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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> 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/>( <span class="p1">(<span class="default"><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>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> )</span><br/><b>let </b><font color="Maroon" title="c2">f</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 ( <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <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"><span class="p3">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> ) )</span><br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E8">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"> ( <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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <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"><span class="p3">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> )</span><br/>

<b>set </b><font color="Maroon" title="c3">F</font> =  <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>;<br/>
<a NAME="E2:9"/>
 <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="matrixj1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="matrixj1.html#D1" target="_self" title="MATRIXJ1:def.1">Def1</a></i>;<br/>
<a NAME="E3:9"/><b>then </b>
1 <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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:9"/><i><font color="Green" title="E9">A2</font></i>: 
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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"><span class="p3">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font></span><br/>

<b>assume </b><a NAME="E5:9"/>
 <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"><span class="p3">(<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:9"/><b>then </b>
<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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrixj1.html#E1:9"><i><font color="Green" title="E8">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="E7:9"/>
contradiction
 <b>by </b><i><a class="txt" href="matrixj1.html#E4:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</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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