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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</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">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Olive" title="b1">L</font>  ex <font color="Olive" title="b3">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Olive" title="b1">L</font> st <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b4">XY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b5">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b4">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b4">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b5">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b3">q</font>,<font color="Olive" title="b5">XanyY</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">L</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</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">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  ex <font color="Olive" title="b2">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font> st <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b3">XY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font><br/>  for <font color="Olive" title="b4">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Olive" title="b3">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b3">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b4">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">XanyY</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">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">  ex <font color="Olive" title="b1">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font> st <br/>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b2">XY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font><br/>  for <font color="Olive" title="b3">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Olive" title="b2">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b2">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b3">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b2">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">XanyY</font>) ) )</span><br/>









<span class="kw">consider </span><font color="Maroon" title="c6">P</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c4">L</font><span class="kw"> such that </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font><br/>  for <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Olive" title="b2">y</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c6">P</font>,<font color="Olive" title="b2">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_2.html#T27" target="_self" onmouseover="rs('hilb10_2/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P1</font> = <font color="Maroon" title="c6">P</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">I</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>;<br/>
<a NAME="E5:50"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">I</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T5" onmouseover="rs('afinsq_1/T5')" onmouseout="rh()">AFINSQ_1:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">nm</font> = <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Inm</font> = <font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>;<br/>
<a NAME="E7:50"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:50"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
;<br/>
<a NAME="E9:50"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="afinsq_1.html#T26" onmouseover="rs('afinsq_1/T26')" onmouseout="rh()">AFINSQ_1:26</a></span>;<br/>
<a NAME="E10:50"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="hilb10_2.html#T1" target="_self" onmouseover="rs('hilb10_2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E11:50"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T26" onmouseover="rs('afinsq_1/T26')" onmouseout="rh()">AFINSQ_1:26</a></span>;<br/>
<a NAME="E12:50"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="hilb10_2.html#T1" target="_self" onmouseover="rs('hilb10_2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E13:50"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E14:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T52" onmouseover="rs('card_fin/T52')" onmouseout="rh()">CARD_FIN:52</a></span>;<br/>
<a NAME="E15:50"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T26" onmouseover="rs('afinsq_1/T26')" onmouseout="rh()">AFINSQ_1:26</a></span>;<br/>
<a NAME="E16:50"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E17:50"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:50"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xboole_1.html#T70" onmouseover="rs('xboole_1/T70')" onmouseout="rh()">XBOOLE_1:70</a></span>;<br/>
<a NAME="E18:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="card_fin.html#T52" onmouseover="rs('card_fin/T52')" onmouseout="rh()">CARD_FIN:52</a></span>;<br/>
<span class="lab"><font color="Green" title="E41">A12</font></span>:  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T26" onmouseover="rs('afinsq_1/T26')" onmouseout="rh()">AFINSQ_1:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T26" onmouseover="rs('afinsq_1/T26')" onmouseout="rh()">AFINSQ_1:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
;<br/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">III</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<a NAME="E22:50"/>
<font color="Maroon" title="c12">III</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">III</font> = <font color="Maroon" title="c12">III</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:50"><span class="lab"><font color="Green" title="E40">A11</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c14">T</font> = <font color="Maroon" title="c7">P1</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">III</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; (  for <font color="Olive" title="b1">XY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font><br/>  for <font color="Olive" title="b2">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Olive" title="b1">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b1">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b2">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Olive" title="b2">XanyY</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E24:50"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="hilb10_2.html#T26" target="_self" onmouseover="rs('hilb10_2/T26')" onmouseout="rh()">Th26</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">  for <font color="Olive" title="b1">XY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font><br/>  for <font color="Olive" title="b2">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Olive" title="b1">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b1">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b2">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Olive" title="b2">XanyY</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">XY</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">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">  for <font color="Olive" title="b1">XanyY</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font>  st <font color="Maroon" title="c15">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Olive" title="b1">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> holds <br/> <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c15">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Olive" title="b1">XanyY</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c16">XanyY</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">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="c15">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> implies  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c15">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Maroon" title="c16">XanyY</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E25:50"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
( <font color="Maroon" title="c15">XY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XanyY</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</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">  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c15">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Maroon" title="c16">XanyY</font>)</span><br/>

<a NAME="E26:50"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> )
 
<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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
;<br/>
<span class="lab"><font color="Green" title="E45">A16</font></span>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
;<br/>
<a NAME="E29:50"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="afinsq_1.html#T54" onmouseover="rs('afinsq_1/T54')" onmouseout="rh()">AFINSQ_1:54</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E46">A17</font></span>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
;<br/>
<a NAME="E31:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A18</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>;<br/>
<a NAME="E32:50"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">R</font> = <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c4">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:50"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">r</font> = <font color="Maroon" title="c17">R</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c4">L</font> ;<br/>
<font color="Maroon" title="c17">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T58" onmouseover="rs('afinsq_1/T58')" onmouseout="rh()">AFINSQ_1:58</a>, <a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a></span>
<br/>.= 
 <a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font>

;<br/>
<a NAME="E36:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A19</font></span>: 
 <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c15">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c17">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/>
<a NAME="E37:50"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">III</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">III</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T43" onmouseover="rs('funct_1/T43')" onmouseout="rh()">FUNCT_1:43</a></span>;<br/>
<a NAME="E38:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A20</font></span>: 
 <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c18">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_2.html#T25" target="_self" onmouseover="rs('hilb10_2/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E39:50"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</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/>
<a NAME="E40:50"/>
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E41:50"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">I</font>
 
;<br/>
<a NAME="E42:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A22</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T54" onmouseover="rs('afinsq_1/T54')" onmouseout="rh()">AFINSQ_1:54</a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>;<br/>
<a NAME="E43:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/>
<a NAME="E44:50"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E45:50"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:50"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="afinsq_1.html#T54" onmouseover="rs('afinsq_1/T54')" onmouseout="rh()">AFINSQ_1:54</a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>;<br/>
<a NAME="E46:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A26</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/>
<a NAME="E47:50"/><span class="lab"><font color="Green" title="E56">A27</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:50"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T17" onmouseover="rs('afinsq_1/T17')" onmouseout="rh()">AFINSQ_1:17</a></span>;<br/>
<a NAME="E48:50"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">w</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="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> implies <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_7"/><span class="lab"><font color="Green" title="E57">A28</font></span>: 
<font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</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"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_7_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="E1:50_7_1"/>
( <font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> or ( <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">w</font> &amp; <font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> ) or <font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_7_1_1"/><span class="lab"><font color="Green" title="E58">A29</font></span>: 
<font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</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"> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font></span><br/><div class="add"><a NAME="E2:50_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A30</font></span>: 
( <font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a>, <a class="ref" href="afinsq_1.html#T21" onmouseover="rs('afinsq_1/T21')" onmouseout="rh()">AFINSQ_1:21</a></span>;<br/><a NAME="E3:50_7_1_1"/>
<font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E44:50"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:50_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A31</font></span>: 
<font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><span class="lab"><font color="Green" title="E61">A32</font></span>: <font color="Maroon" title="c13">III</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a>, <a class="txt" href="#E2:50_7_1_1"><span class="lab"><font color="Green" title="E59">A30</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_7_1_1"><span class="lab"><font color="Green" title="E59">A30</font></span></a>, <a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c9">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_7_1_1"><span class="lab"><font color="Green" title="E60">A31</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E1:50_7"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="funct_1.html#T17" onmouseover="rs('funct_1/T17')" onmouseout="rh()">FUNCT_1:17</a></span>
;<br/><a NAME="E6:50_7_1_1"/>
<font color="Maroon" title="c19">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:50_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> = 
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c16">XanyY</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:50"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E1:50_7_1_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:50_7_1_1"/>
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_7_1_1"><span class="lab"><font color="Green" title="E61">A32</font></span></a>, <a class="txt" href="#E1:50_7"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E50">A21</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_7_1_2"/><span class="lab"><font color="Green" title="E58">A33</font></span>: 
( <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c19">w</font> &amp; <font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</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"> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">wn</font> = <font color="Maroon" title="c19">w</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/><a NAME="E3:50_7_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a></span>;<br/><a NAME="E4:50_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A34</font></span>: 
<font color="Maroon" title="c20">wn</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:50_7_1_2"/><span class="lab"><font color="Green" title="E60">A35</font></span>: 
<font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><a NAME="E6:50_7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">w</font>
 ;<br/><a NAME="E7:50_7_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:50_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><a NAME="E9:50_7_1_2"/>
<font color="Maroon" title="c19">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a>, <a class="txt" href="#E47:50"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A37</font></span>: <font color="Maroon" title="c13">III</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">wn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7_1_2"><span class="lab"><font color="Green" title="E58">A33</font></span></a>, <a class="ref" href="afinsq_1.html#T18" onmouseover="rs('afinsq_1/T18')" onmouseout="rh()">AFINSQ_1:18</a>, <a class="txt" href="#E43:50"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:50"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E4:50_7_1_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_7_1_2"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="ref" href="funct_1.html#T17" onmouseover="rs('funct_1/T17')" onmouseout="rh()">FUNCT_1:17</a></span>
;<br/><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wn</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">wn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E30:50"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E4:50_7_1_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">wn</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:50"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E4:50_7_1_2"><span class="lab"><font color="Green" title="E59">A34</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_7_1_2"><span class="lab"><font color="Green" title="E60">A35</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:50_7_1_2"/>
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_7_1_2"><span class="lab"><font color="Green" title="E62">A37</font></span></a>, <a class="txt" href="#E1:50_7"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E50">A21</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_7_1_3"><span class="kw">suppose </span></a><a NAME="E1:50_7_1_3"/>
<font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">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"> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">wnk</font> = <font color="Maroon" title="c19">w</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/><a NAME="E3:50_7_1_3"/><span class="lab"><font color="Green" title="E58">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_7"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="afinsq_1.html#T66" onmouseover="rs('afinsq_1/T66')" onmouseout="rh()">AFINSQ_1:66</a></span>;<br/><a NAME="E4:50_7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A39</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">wnk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:50_7_1_3"/><span class="lab"><font color="Green" title="E60">A40</font></span>: 
<font color="Maroon" title="c20">wnk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a>, <a class="txt" href="#E3:50_7_1_3"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E6:50_7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A41</font></span>: 
<font color="Maroon" title="c20">wnk</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:50"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/><a NAME="E7:50_7_1_3"/>
<font color="Maroon" title="c20">wnk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_7_1_3"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:50_7_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">wnk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E9:50_7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A42</font></span>: 
( <font color="Maroon" title="c20">wnk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c20">wnk</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_7_1_3"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><span class="lab"><font color="Green" title="E63">A43</font></span>: <font color="Maroon" title="c13">III</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">wnk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_7_1_3"><span class="lab"><font color="Green" title="E60">A40</font></span></a>, <a class="txt" href="#E46:50"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E47:50"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_7_1_3"><span class="lab"><font color="Green" title="E61">A41</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c9">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_7_1_3"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_7_1_3"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="ref" href="funct_1.html#T17" onmouseover="rs('funct_1/T17')" onmouseout="rh()">FUNCT_1:17</a></span>
;<br/><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E9:50_7_1_3"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="ref" href="afinsq_1.html#D3" onmouseover="rs('afinsq_1/D3')" onmouseout="rh()">AFINSQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c15">XY</font></span>)</span> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">wnk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_7_1_3"><span class="lab"><font color="Green" title="E60">A40</font></span></a>, <a class="txt" href="#E27:50"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c20">wnk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:50"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E28:50"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E5:50_7_1_3"><span class="lab"><font color="Green" title="E60">A40</font></span></a>, <a class="ref" href="afinsq_2.html#D2" onmouseover="rs('afinsq_2/D2')" onmouseout="rh()">AFINSQ_2:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:50_7_1_3"/>
<span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">III</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hilb10_2.html#K2" title="HILB10_2:func.2">@</a> <font color="Maroon" title="c16">XanyY</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_7_1_3"><span class="lab"><font color="Green" title="E63">A43</font></span></a>, <a class="txt" href="#E1:50_7"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E50">A21</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E49:50"/>
 <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c15">XY</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K2" title="POLYNOM2:func.2">eval</a> (<font color="Maroon" title="c14">T</font>,<font color="Maroon" title="c16">XanyY</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:50"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E36:50"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E39:50"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="afinsq_1.html#T8" onmouseover="rs('afinsq_1/T8')" onmouseout="rh()">AFINSQ_1:8</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>
