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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 ) )</span><br/>

<b>thus </b><a NAME="E1:62"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 )</span><br/>

<a NAME="E2:62_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_3.html#V11" title="BCIALG_3:attr.11">BCK-implicative</a> <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T50" target="_self" title="BCIALG_5:th.50">Th47</a></i>;<br/>
<b>hence </b><a NAME="E3:62_1"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 )
 <b>by </b><i><a class="ref" href="bcialg_5.html#T49" target="_self" title="BCIALG_5:th.49">Th46</a>, <a class="ref" href="bcialg_5.html#T38" target="_self" title="BCIALG_5:th.38">Th41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:62"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_3.html#V1" title="BCIALG_3:attr.1">commutative</a> <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 
<b>by </b><i><a class="ref" href="bcialg_5.html#T38" target="_self" title="BCIALG_5:th.38">Th41</a></i>;<br/>
<a NAME="E4:62"/>
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_3.html#V10" title="BCIALG_3:attr.10">BCK-positive-implicative</a> <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="bcialg_5.html#T49" target="_self" title="BCIALG_5:th.49">Th46</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
<font color="Maroon" title="c1">X</font> is  <a href="bcialg_3.html#V11" title="BCIALG_3:attr.11">BCK-implicative</a> <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 
<b>by </b><i><a class="txt" href="#E3:62"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="bcialg_3.html#T34" title="BCIALG_3:th.34">BCIALG_3:34</a></i>;<br/>
<b>hence </b><a NAME="E6:62"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="bcialg_5.html#T50" target="_self" title="BCIALG_5:th.50">Th47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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