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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>, <font color="Olive" title="b2">y</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="b3">a</font>, <font color="Olive" title="b4">b</font> being    <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>  st <font color="Olive" title="b3">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b4">b</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <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="b2">y</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Olive" title="b3">a</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Olive" title="b4">b</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">y</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="Olive" title="b4">b</font> 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="b3">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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">a</font>, <font color="Olive" title="b2">b</font> being    <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>  st <font color="Olive" title="b1">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b2">b</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <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="c3">y</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Olive" title="b2">b</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c3">y</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="Olive" title="b2">b</font> 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">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <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>; <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="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <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="c3">y</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Maroon" title="c4">a</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a>  &amp; <font color="Maroon" title="c5">b</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="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="c4">a</font> &amp; <font color="Maroon" title="c3">y</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> implies  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <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">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E29">A2</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="E3:39"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c3">y</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 
 <span class="kw">and </span><br/><a NAME="E4:39"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c4">a</font> is  <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 
 <span class="kw">and </span><br/><a NAME="E5:39"/><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">and </span><br/><a NAME="E6:39"/><span class="lab"><font color="Green" title="E33">A6</font></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="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c3">y</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>
 ; <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="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<a NAME="E8:39"/>
 <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D4" onmouseover="rs('nat_d/D4')" onmouseout="rh()">NAT_D:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">yx</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:39"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">yx</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="c7">na</font> =  <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c4">a</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">xly</font> = <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> 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="E13:39"/>
 <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D4" onmouseover="rs('nat_d/D4')" onmouseout="rh()">NAT_D:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">xy</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:39"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">xy</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="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">xly</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</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"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">xy</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">yx</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="bcialg_6.html#T15" target="_self" onmouseover="rs('bcialg_6/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">yx</font></span>)</span></span>)</span>
<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="c4">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<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="c4">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c7">na</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="bcialg_6.html#T28" target="_self" onmouseover="rs('bcialg_6/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="bcialg_6.html#T26" target="_self" onmouseover="rs('bcialg_6/T26')" onmouseout="rh()">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <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> <span class="p3">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="bcialg_6.html#T28" target="_self" onmouseover="rs('bcialg_6/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c9">xy</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<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="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="bcialg_6.html#T26" target="_self" onmouseover="rs('bcialg_6/T26')" onmouseout="rh()">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c6">yx</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#T7" target="_self" onmouseover="rs('bcialg_6/T7')" onmouseout="rh()">Th7</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_1.html#K2" title="BCIALG_1:func.2">`</a> 
<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 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#D5" onmouseover="rs('bcialg_1/D5')" onmouseout="rh()">BCIALG_1:def 5</a></span>
;<br/>
<a NAME="E17:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">xly</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_1.html#D5" onmouseover="rs('bcialg_1/D5')" onmouseout="rh()">BCIALG_1:def 5</a></span>;<br/>
<a NAME="E18:39"/><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="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">xly</font>
 
;<br/>
<a NAME="E19:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c8">xly</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/>
<span class="kw">hence </span><a NAME="E20:39"/>
 <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="nat_d.html#K5" title="NAT_D:func.5">lcm</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K3" title="BCIALG_6:func.3">ord</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="bcialg_6.html#T29" target="_self" onmouseover="rs('bcialg_6/T29')" onmouseout="rh()">Th29</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/>


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