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<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> )
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">r</font> =  <a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/>
<a NAME="E4:25"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1"/><i><font color="Green">E48</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:25_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T2">FINSEQ_2:2</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><a NAME="E5:25_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:25_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">r</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:25"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E7:25"/>
verum
 ;<br/>


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