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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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b4">n</font> iff <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Olive" title="b3">m</font>,<font color="Olive" title="b4">n</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">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="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font> iff <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> )</span><br/>









<span class="kw">thus </span><a NAME="E1:22"/>
( <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font> implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</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" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</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" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span><br/>

<a NAME="E2:22_1"/><span class="kw">then </span>
 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> (<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<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> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)
 
<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>;<br/>
<span class="kw">hence </span><a NAME="E3:22_1"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:22"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</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" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E3:22"/><span class="kw">then </span>
 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> (<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 
<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>;<br/>
<span class="kw">hence </span><a NAME="E4:22"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_5.html#NM2" title="BCIALG_5:NM.2">BCK-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <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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