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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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#K4" title="BCIALG_6:func.4">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#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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#K4" title="BCIALG_6:func.4">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#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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#K4" title="BCIALG_6:func.4">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#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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="E1:41"/><span class="lab"><font color="Green" title="E29">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="E2:41"/><span class="lab"><font color="Green" title="E30">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="E3:41"/><span class="lab"><font color="Green" title="E31">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="E4:41"/><span class="lab"><font color="Green" title="E32">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="E5:41"/><span class="lab"><font color="Green" title="E33">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="E6:41"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN: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="E7:41"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN: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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<a NAME="E8:41"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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" title="NAT_D:def.4">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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:41"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">yx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">na</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">nb</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">xly</font> = <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E13:41"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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" title="NAT_D:def.4">NAT_D:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">xy</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:41"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">xy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c9">xly</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">yx</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E10:41"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="bcialg_6.html#T15" target="_self" title="BCIALG_6:th.15">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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" title="BCIALG_6:th.23">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</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" title="BCIALG_6:th.23">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c7">na</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="bcialg_6.html#T28" target="_self" title="BCIALG_6:th.28">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="bcialg_6.html#T26" target="_self" title="BCIALG_6:th.26">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="bcialg_6.html#T28" target="_self" title="BCIALG_6:th.28">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c10">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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c6">yx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="bcialg_6.html#T26" target="_self" title="BCIALG_6:th.26">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#K3" title="BCIALG_6:func.3">|^</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" title="BCIALG_6:th.7">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" title="BCIALG_6:th.7">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" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>
;<br/>
<a NAME="E17:41"/><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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c9">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" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>;<br/>
<a NAME="E18:41"/><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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c9">xly</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E19:41"/><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#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c9">xly</font> <a href="hidden.html#R2" title="HIDDEN: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="E20:41"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="bcialg_6.html#T29" target="_self" title="BCIALG_6:th.29">Th29</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">nab</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">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> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>

</div>
