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<span class="kw">let </span><font color="Maroon" title="c3">IT</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="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> iff ( <font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">IT</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:19_1_1"/>
( <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> implies ( <font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">IT</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>; <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="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">IT</font> ) implies <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E3:19_1_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c3">IT</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:19_1_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">IT</font>
 ; <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="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E5:19_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">IT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">m</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">IT</font> )</span><br/>

<a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i</font> being    <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"> such that </span><br/><a NAME="E3:19_1_1_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:19_1_1_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E5:19_1_1_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">n</font>
 ; <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="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">IT</font></span><br/>

<a NAME="E6:19_1_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></span>;<br/>
<a NAME="E7:19_1_1_1"/>
<font color="Maroon" title="c5">i</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="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E5:19_1_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></span>;<br/>
<a NAME="E8:19_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">IT</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:19_1_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:19_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:19_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</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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