<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">m1</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font>, <font color="Maroon">m2</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">p</font> = <font color="Maroon">m1</font> <a href="polynom8.html#K2" target="_self">*</a> <font color="Maroon">m2</font>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m2</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">k</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m2</font></span>)</span> 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="square_1.html#T38">SQUARE_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">m1</font> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">m2</font> )
 </span><b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">m1</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1"/><b>then </b>
<font color="Maroon">m1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><b>hence </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom8.html#K2" target="_self">*</a> <font color="Maroon">m2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">m2</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2"/><b>then </b>
<font color="Maroon">m2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/><b>hence </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom8.html#K2" target="_self">*</a> <font color="Maroon">m2</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:17"/><b>then </b>
<font color="Maroon">k</font> <a href="algseq_1.html#R1">is_at_least_length_of</a> <font color="Maroon">m1</font> <a href="polynom8.html#K2" target="_self">*</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:17"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="polynom8.html#K2" target="_self">*</a> <font color="Maroon">m2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">m2</font></span>)</span>
 <b>by </b><i><a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


</div>
