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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">x</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b2">m</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp;  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Olive" title="b3">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">x</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b1">m</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp;  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Olive" title="b2">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp;  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c2">x</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 
 <span class="kw">and </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E4:36"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T34" onmouseover="rs('bcialg_1/T34')" onmouseout="rh()">BCIALG_1:34</a></span>;<br/>
<a NAME="E6:36"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c5">b</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="bcialg_6.html#T25" target="_self" onmouseover="rs('bcialg_6/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E7:36"/>
<font color="Maroon" title="c5">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T1" onmouseover="rs('bcialg_1/T1')" onmouseout="rh()">BCIALG_1:1</a></span>;<br/>
<a NAME="E8:36"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
;<br/>
<a NAME="E9:36"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c5">b</font>
 
;<br/>
<a NAME="E10:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c5">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="bcialg_6.html#T28" target="_self" onmouseover="rs('bcialg_6/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E11:36"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="bcialg_6.html#D5" target="_self" onmouseover="rs('bcialg_6/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E12:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> )
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">mgn</font> = <font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E14:36"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T21" onmouseover="rs('int_2/T21')" onmouseout="rh()">INT_2:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">vn</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:36"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">mgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">vn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">vn</font> = <font color="Maroon" title="c7">vn</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E18:36"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T21" onmouseover="rs('int_2/T21')" onmouseout="rh()">INT_2:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E20:36"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">mgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E22:36"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<a NAME="E23:36"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c8">vn</font>,<font color="Maroon" title="c10">i</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:36_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a></span>;<br/>
<a NAME="E5:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E6:36_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c8">vn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">b1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:36_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<font color="Maroon" title="c8">vn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E9:36_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E8:36_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a></span>;<br/>
<a NAME="E10:36_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E11:36_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:36_1"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E14:36_1"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="int_2.html#T5" onmouseover="rs('int_2/T5')" onmouseout="rh()">INT_2:5</a></span>;<br/>
<a NAME="E15:36_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>;<br/>
<a NAME="E16:36_1"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c8">vn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:36_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="nat_1.html#T15" onmouseover="rs('nat_1/T15')" onmouseout="rh()">NAT_1:15</a>, <a class="ref" href="xcmplx_1.html#T5" onmouseover="rs('xcmplx_1/T5')" onmouseout="rh()">XCMPLX_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:36_1"/>
<font color="Maroon" title="c8">vn</font>,<font color="Maroon" title="c10">i</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" onmouseover="rs('int_2/D3')" onmouseout="rh()">INT_2:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:36"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
 for <font color="Olive" title="b1">mm</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b1">mm</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon" title="c8">vn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">mm</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c11">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c8">vn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">mm</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_2"/><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E2:36_2"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<font color="Maroon" title="c11">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">vn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">mm</font></span><br/>

<a NAME="E3:36_2"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_2"><span class="lab"><font color="Green" title="E40">A18</font></span></a></span>;<br/>
<a NAME="E4:36_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E5:36_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T18" target="_self" onmouseover="rs('bcialg_6/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E6:36_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T21" target="_self" onmouseover="rs('bcialg_6/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E7:36_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c11">mm</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T18" target="_self" onmouseover="rs('bcialg_6/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E8:36_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">mm</font></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T23" target="_self" onmouseover="rs('bcialg_6/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E9:36_2"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">mm</font></span>)</span>
 
;<br/>
<a NAME="E10:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">mm</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E11:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">vn</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">mm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="bcialg_6.html#T29" target="_self" onmouseover="rs('bcialg_6/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:36_2"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E14:36_2"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<font color="Maroon" title="c6">mgn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="int_2.html#T5" onmouseover="rs('int_2/T5')" onmouseout="rh()">INT_2:5</a></span>;<br/>
<a NAME="E15:36_2"/>
<font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">mm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">mgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_2"><span class="lab"><font color="Green" title="E42">A20</font></span></a></span>;<br/>
<a NAME="E16:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:36_2"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" onmouseover="rs('xcmplx_1/T5')" onmouseout="rh()">XCMPLX_1:5</a></span>;<br/>
<a NAME="E17:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">vn</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c11">mm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E18:36_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">vn</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c11">mm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:36"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="int_2.html#T25" onmouseover="rs('int_2/T25')" onmouseout="rh()">INT_2:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">cc</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E20:36_2"/><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<font color="Maroon" title="c11">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">cc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/>
<a NAME="E21:36_2"/>
<font color="Maroon" title="c13">cc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_2"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E20:36_2"><span class="lab"><font color="Green" title="E44">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:36_2"/>
<font color="Maroon" title="c8">vn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_2"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="nat_1.html#T24" onmouseover="rs('nat_1/T24')" onmouseout="rh()">NAT_1:24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T18" target="_self" onmouseover="rs('bcialg_6/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T21" target="_self" onmouseover="rs('bcialg_6/T21')" onmouseout="rh()">Th21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T18" target="_self" onmouseover="rs('bcialg_6/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">mgn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">vn</font></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="bcialg_6.html#T23" target="_self" onmouseover="rs('bcialg_6/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T23" target="_self" onmouseover="rs('bcialg_6/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T18" target="_self" onmouseover="rs('bcialg_6/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c10">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T7" target="_self" onmouseover="rs('bcialg_6/T7')" onmouseout="rh()">Th7</a></span>
;<br/>
<a NAME="E26:36"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font>
 
;<br/>
<a NAME="E27:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">vn</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E28:36"/>
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="newton.html#T58" onmouseover="rs('newton/T58')" onmouseout="rh()">NEWTON:58</a></span>;<br/>
<a NAME="E29:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c6">mgn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:36"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="nat_d.html#D1" onmouseover="rs('nat_d/D1')" onmouseout="rh()">NAT_D:def 1</a></span>;<br/>
<a NAME="E30:36"/>
<font color="Maroon" title="c8">vn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E16:36"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="bcialg_6.html#D5" target="_self" onmouseover="rs('bcialg_6/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:36"/>
 <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E29:36"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="txt" href="#E27:36"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E24:36"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="bcialg_6.html#D5" target="_self" onmouseover="rs('bcialg_6/D5')" onmouseout="rh()">Def5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
