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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="E1:22"/><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">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="E2:22"/><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>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:22"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</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>
 ;<br/>

<a NAME="E4:22"/><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="E5:22"/>
 <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="E6:22"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <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>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T2">FINSEQ_2:2</a></i>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">j</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">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font>, <font color="Maroon">j1</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E9:22"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> implies <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="funct_1.html#K1">.</a> <font color="Maroon">j</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="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>thus </b><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="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E10:22"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E11:22"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>thus </b><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="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i>, , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>
;<br/>


</div>
