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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="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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b3">n</font></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" 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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> $1</span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E14">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" 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="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" 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="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</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#K3" title="BCIALG_6:func.3">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c5">n</font></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#T2" target="_self" title="BCIALG_6:th.2">Th2</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_6.html#E18"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E3:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D1" target="_self" title="BCIALG_6:def.1">Def1</a></span>;<br/>
<a NAME="E3:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D1" target="_self" title="BCIALG_6:def.1">Def1</a></span>;<br/>
<a NAME="E4:19"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E14">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 ; <a class="txt" 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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