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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E13">Th11</font></span>: <a NAME="T12"><span class="comment"><font color="firebrick">:: HILB10_5:12</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">n1</font>, <font color="Olive" title="b3">n2</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font>, <font color="Olive" title="b6">l</font>, <font color="Olive" title="b7">m</font>, <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b9">i1</font>, <font color="Olive" title="b10">i2</font>, <font color="Olive" title="b11">i3</font>, <font color="Olive" title="b12">i4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b8">n</font>  st <font color="Olive" title="b2">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">n</font> holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b13">p</font> where <font color="Olive" title="b13">p</font> is  <font color="Olive">b<sub>8</sub></font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Olive" title="b13">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b9">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b13">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b10">i2</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="hilb10_4.html#K1" title="HILB10_4:func.1">Product</a> <span class="p5">(<span class="default">1 <a href="valued_1.html#K7" title="VALUED_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Olive" title="b13">p</font> <a href="afinsq_2.html#K2" title="AFINSQ_2:func.2">/^</a> <font color="Olive" title="b2">n1</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b3">n2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b6">l</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Olive" title="b13">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b11">i3</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b7">m</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b13">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b12">i4</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> </span> } </span>  is   <a href="hilb10_2.html#V1" title="HILB10_2:attr.1">diophantine</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b8">n</font> <a href="hilb10_2.html#K5" title="HILB10_2:func.5">-xtuples_of</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></div></div>
