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<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="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#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> holds   <a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font></span><br/>



<div><span class="lab"><font color="Green" title="E13">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> implies <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:12_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D10" title="BHSP_1:def.10">BHSP_1:def 10</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="bhsp_1.html#D10" title="BHSP_1:def.10">BHSP_1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> = 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T11" title="POLYNOM4:th.11">POLYNOM4:11</a></span>
<br/>.= 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T11" title="POLYNOM4:th.11">POLYNOM4:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:12"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="algseq_1.html#T28" title="ALGSEQ_1:th.28">ALGSEQ_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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