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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="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</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="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></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="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">P</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<div><span class="lab"><font color="Green" title="E11">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 implies <font color="Maroon" title="c3">P</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</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><br/><span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <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">P</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</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><br/><span class="kw">hence </span><font color="Maroon" title="c3">P</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>
<br/>.= 
 <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="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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 NAME="E3:17"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E4:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></span>;<br/>
<a NAME="E5:17"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17"/>
 <a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="polynom4.html#D1" target="_self" title="POLYNOM4:def.1">Def1</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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