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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"> 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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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#M1" title="BCIALG_5:mode.1">BCI-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/><b>let </b><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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#M1" title="BCIALG_5:mode.1">BCI-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#M1" title="BCIALG_5:mode.1">BCI-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/>









<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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#M1" title="BCIALG_5:mode.1">BCI-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" 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#M1" title="BCIALG_5:mode.1">BCI-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#M1" title="BCIALG_5:mode.1">BCI-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" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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" 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#M1" title="BCIALG_5:mode.1">BCI-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="E2:18_1"/>
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>
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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>
 <b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</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="E2:18"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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" 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#M1" title="BCIALG_5:mode.1">BCI-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="E3:18"/>
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>
 
<b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-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>
 <b>by </b><i><a class="ref" href="bcialg_5.html#D3" target="_self" title="BCIALG_5:def.3">Def2</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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