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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>



<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" title="NAT_D:th.67">NAT_D:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>

;<br/>
<span class="kw">hence </span><a NAME="E2:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>; <a class="txt" 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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