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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">M</font> = <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">N</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">L</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E1:74"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E2:74"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E3:74"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E4:74"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E5:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E6:74"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E7:74"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E8:74"/>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E9:74"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E10:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:74"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E11:74"/>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E12:74"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E13:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:74"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E14:74"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E15:74"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E16:74"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:74"/>
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:74"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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