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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">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">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 8 implies  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:114"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 8
 ; <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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_1"/>
( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="number06.html#T18" onmouseover="rs('number06/T18')" onmouseout="rh()">NUMBER06:18</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1"/>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><a NAME="E2:114_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c1">k</font> is    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#D15" target="_self" onmouseover="rs('number14/D15')" onmouseout="rh()">Def15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_1_1_1"/>
( <font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_1_1_1"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_1_1_1"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_1_1"><span class="lab"><font color="Green" title="E82">A4</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><a NAME="E7:114_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">hence </span><a NAME="E8:114_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_1_1"><span class="lab"><font color="Green" title="E81">A3</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_1_1_2"/><span class="lab"><font color="Green" title="E81">A5</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_1"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_1_1_2"/><span class="lab"><font color="Green" title="E82">A6</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_1_1_2"/>
6 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_1_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_1_1_2"><span class="lab"><font color="Green" title="E82">A6</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><span class="kw">thus </span><a NAME="E7:114_1_1_1_2"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1_1_2"><span class="lab"><font color="Green" title="E81">A5</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2"/>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><a NAME="E2:114_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A7</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#D15" target="_self" onmouseover="rs('number14/D15')" onmouseout="rh()">Def15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_1_2_1"/>
( <font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_2"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_2_1_1"/><span class="lab"><font color="Green" title="E82">A9</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 6
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_2_1_1"/>
7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_2_1_1"><span class="lab"><font color="Green" title="E82">A9</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><span class="kw">thus </span><a NAME="E7:114_1_2_1_1"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_2_1_1"><span class="lab"><font color="Green" title="E81">A8</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_2_1_2"/><span class="lab"><font color="Green" title="E81">A10</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_2"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_2_1_2"/><span class="lab"><font color="Green" title="E82">A11</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_2_1_2"/>
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_2_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_2_1_2"><span class="lab"><font color="Green" title="E82">A11</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><a NAME="E7:114_1_2_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E8:114_1_2_1_2"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_2_1_2"><span class="lab"><font color="Green" title="E81">A10</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_3"><span class="kw">suppose </span></a><a NAME="E1:114_1_3"/>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><a NAME="E2:114_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A12</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 is    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#D15" target="_self" onmouseover="rs('number14/D15')" onmouseout="rh()">Def15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:114_1_3_1"/>
( <font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:114_1_3_1_1"/><span class="lab"><font color="Green" title="E81">A13</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_3"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_3_1_1"/><span class="lab"><font color="Green" title="E82">A14</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_3_1_1"/>
5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:114_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:114_1_3_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_3_1_1"><span class="lab"><font color="Green" title="E82">A14</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><a NAME="E7:114_1_3_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">hence </span><a NAME="E8:114_1_3_1_1"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_3_1_1"><span class="lab"><font color="Green" title="E81">A13</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:114_1_3_1_2"/><span class="lab"><font color="Green" title="E81">A15</font></span>: 
<font color="Maroon" title="c1">k</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">  ex <font color="Olive" title="b1">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Olive" title="b1">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1_3"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="number14.html#T58" target="_self" onmouseover="rs('number14/T58')" onmouseout="rh()">Th58</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">m</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="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/><a NAME="E3:114_1_3_1_2"/><span class="lab"><font color="Green" title="E82">A16</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 7
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/><a NAME="E4:114_1_3_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:114_1_3_1_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:114_1_3_1_2"><span class="lab"><font color="Green" title="E82">A16</font></span></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"> <font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> </span><br/><span class="kw">thus </span><a NAME="E6:114_1_3_1_2"/>
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_3_1_2"><span class="lab"><font color="Green" title="E81">A15</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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