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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_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of 1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:60"/>
( <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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> ( <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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</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#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1 )</span><br/>

<a NAME="E2:60_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T50" target="_self" onmouseover="rs('bcialg_5/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:60_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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T38" target="_self" onmouseover="rs('bcialg_5/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="bcialg_5.html#T49" target="_self" onmouseover="rs('bcialg_5/T49')" onmouseout="rh()">Th49</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>

<span class="kw">assume </span><a NAME="E2:60"/>
( <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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a> ,1, <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E3:60"/><span class="kw">then </span>
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T38" target="_self" onmouseover="rs('bcialg_5/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="bcialg_5.html#T49" target="_self" onmouseover="rs('bcialg_5/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E4:60"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_3.html#T34" onmouseover="rs('bcialg_3/T34')" onmouseout="rh()">BCIALG_3:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:60"/>
<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#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_5.html#T50" target="_self" onmouseover="rs('bcialg_5/T50')" onmouseout="rh()">Th50</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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