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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"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#V15" title="BCIALG_1:attr.15">positive-implicative</a>  <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a> implies <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1,1,1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#V15" title="BCIALG_1:attr.15">positive-implicative</a>  <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"> <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1,1,1</span><br/>

<a NAME="E2:54"/>
 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> holds   <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (1,1,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (1,1,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>)</span><br/>

<a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</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="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bcialg_1.html#T85" onmouseover="rs('bcialg_1/T85')" onmouseout="rh()">BCIALG_1:85</a></span>;<br/>
(<span class="p1">(<span class="default">(<font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1 = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1</span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T2" onmouseover="rs('bcialg_2/T2')" onmouseout="rh()">BCIALG_2:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T2" onmouseover="rs('bcialg_2/T2')" onmouseout="rh()">BCIALG_2:2</a></span>
<br/>.= 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="bcialg_2.html#T2" onmouseover="rs('bcialg_2/T2')" onmouseout="rh()">BCIALG_2:2</a></span>
<br/>.= 
(<span class="p1">(<span class="default">(<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T3" onmouseover="rs('bcialg_2/T3')" onmouseout="rh()">BCIALG_2:3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:54_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (1,1,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:54"/>
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1,1,1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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