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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be    <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/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E5">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T6" title="EUCLID_2:th.6">EUCLID_2:6</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp;  <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> )
 
<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="E5:11"/><i><font color="Green" title="E9">A5</font></i>: 
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E3:11"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green" title="E10">A6</font></i>: 
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E2:11"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="rfinseq2.html#E4:11"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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="c1">f1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/>
<a NAME="E9:11"/>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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">f2</font>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="rfinseq2.html#E6:11"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:11"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:11"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self" title="RFINSEQ2:def.2">Def2</a></i>;<br/>
<b>thus </b><a NAME="E11:11"/>
 <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E5:11"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="rfinseq2.html#E8:11"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="rfinseq2.html#E10:11"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>


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