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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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"> <span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> </span><br/>

<a NAME="E1:122"/>
( <font color="Maroon" title="c1">D</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or <font color="Maroon" title="c1">D</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E2:122"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E116"><span class="lab"><font color="Green" title="E79">Lm11</font></span></a>, <a class="txt" href="number14.html#E117"><span class="lab"><font color="Green" title="E80">Lm12</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/>


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