<?xml version="1.0"?>
<div><span class="kw">theorem </span><a NAME="T42"><span class="comment"><font color="firebrick">:: RING_3:42</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><br/>  for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font>, <font color="Olive" title="b5">n1</font>, <font color="Olive" title="b6">n2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b6">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b6">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>(  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b6">n2</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">m2</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b6">n2</font></span>)</span></span>)</span> &amp;  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">m2</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">m2</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b6">n2</font></span>)</span></span>)</span> )</div></div>
