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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">r</font> = <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">x</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="c1">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">y</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:37"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:37"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" target="_self" title="NAT_D:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E6:37"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</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" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<a NAME="E7:37"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</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" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<a NAME="E8:37"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="nat_d.html#T11" target="_self" title="NAT_D:th.11">Th11</a></span>;<br/>
<a NAME="E9:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<a NAME="E10:37"/><span class="lab"><font color="Green" title="E25">A5</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</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" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<a NAME="E11:37"/><span class="lab"><font color="Green" title="E26">A6</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<a NAME="E12:37"/>
<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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</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_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c7">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="nat_d.html#T9" target="_self" title="NAT_D:th.9">Th9</a></span>;<br/>
<a NAME="E13:37"/><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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</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="#E3:37"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E11:37"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="nat_d.html#T8" target="_self" title="NAT_D:th.8">Th8</a></span>;<br/>
<a NAME="E14:37"/><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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</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="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="nat_d.html#D5" target="_self" title="NAT_D:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:37"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="nat_d.html#T5" target="_self" title="NAT_D:th.5">Th5</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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