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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">a</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><br/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b2">m</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="Olive" title="b1">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</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">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b1">m</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="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T13" target="_self" onmouseover="rs('bcialg_6/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> $1</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="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E13">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </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">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">m</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T13" target="_self" onmouseover="rs('bcialg_6/T13')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:16_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="bcialg_6.html#T2" target="_self" onmouseover="rs('bcialg_6/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T9" onmouseover="rs('bcialg_1/T9')" onmouseout="rh()">BCIALG_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T29" onmouseover="rs('bcialg_1/T29')" onmouseout="rh()">BCIALG_1:29</a></span>
<br/>.= 
<font color="Maroon" title="c5">p</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T1" target="_self" onmouseover="rs('bcialg_6/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T14" target="_self" onmouseover="rs('bcialg_6/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
<font color="Maroon" title="c7">q</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T1" target="_self" onmouseover="rs('bcialg_6/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E4:16_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D1" target="_self" onmouseover="rs('bcialg_6/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T29" onmouseover="rs('bcialg_1/T29')" onmouseout="rh()">BCIALG_1:29</a></span>
;<br/>
<a NAME="E4:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E5:16"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:16"/>
<font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K2" title="BCIALG_6:func.2">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a></span>)</span>
 ; <a 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>
