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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <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"> ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">fd</font> =  <a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">fm</font> =  <a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">assume </span><a NAME="E1:29"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span><br/>

<a NAME="E2:29"/>
( <span class="p1">(<span class="default"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#D1" target="_self" title="REAL_3:def.1">Def2</a>, <a class="ref" href="real_3.html#D2" target="_self" title="REAL_3:def.2">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:29"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" title="REAL_3:func.2">divSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" title="REAL_3:func.1">modSeq</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</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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