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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Maroon" title="c1">L</font> holds   <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Olive" title="b2">m2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">m2</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m1</font>, <font color="Maroon" title="c3">m2</font> be   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font></span>)</span>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">k</font> =  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font></span>)</span>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">k</font> =  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font></span>)</span>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> holds <br/> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font> implies  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font> or <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" onmouseover="rs('xxreal_0/T15')" onmouseout="rh()">XXREAL_0:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>;<br/><span class="kw">hence </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D7" onmouseover="rs('lopban_3/D7')" onmouseout="rh()">LOPBAN_3:def 7</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:17_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">m2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>;<br/><span class="kw">hence </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D7" onmouseover="rs('lopban_3/D7')" onmouseout="rh()">LOPBAN_3:def 7</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E3:17"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#D2" onmouseover="rs('algseq_1/D2')" onmouseout="rh()">ALGSEQ_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">m2</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#D3" onmouseover="rs('algseq_1/D3')" onmouseout="rh()">ALGSEQ_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
