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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" 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">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</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" 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="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">e</font> =  <a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">o</font> =  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="lab"><font color="Green" title="E12">A1</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> = 
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2"><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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> holds <br/><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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> implies <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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</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"> <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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">x</font> 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:16_2"/>
<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="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T10" target="_self" onmouseover="rs('hurwitz2/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E4:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D2" onmouseover="rs('normsp_1/D2')" onmouseout="rh()">NORMSP_1:def 2</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:16_2"/>
<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T27" onmouseover="rs('polynom3/T27')" onmouseout="rh()">POLYNOM3:27</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:16"/>
<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"><a href="hurwitz2.html#K1" title="HURWITZ2:func.1">even_part</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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/>


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