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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E5</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T65">RVSUM_1:65</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green">E8</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E2:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green">E9</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E10</font></i>:  <a href="rfinseq2.html#K4" target="_self">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<a NAME="E7:13"/><i><font color="Green">E11</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E2:13"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq2.html#E5:13"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E9:13"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<a NAME="E10:13"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<a NAME="E11:13"/><i><font color="Green">E15</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E3:13"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E2:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E13:13"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E9:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rfinseq2.html#E10:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E14:13"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E8:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E15:13"/>
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E3:13"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E16:13"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E2:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E17:13"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E7:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E18:13"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E19:13"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E9:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rfinseq2.html#E10:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rfinseq2.html#E16:13"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E20:13"/><i><font color="Green">E19</font></i>: 
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E3:13"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rfinseq2.html#E4:13"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rfinseq2.html#E14:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E21:13"/>
 <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:13"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E2:13"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq2.html#E11:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rfinseq2.html#E19:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rfinseq2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E22:13"/>
(  <a href="rfinseq2.html#K4" target="_self">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K4" target="_self">min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="rfinseq2.html#K2" target="_self">min_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfinseq2.html#K2" target="_self">min_p</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="rfinseq2.html#E6:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rfinseq2.html#E20:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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