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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_4.html#NM1" title="BCIALG_4:NM.1">BCI-Algebra_with_Condition(S)</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Olive" title="b3">z</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/>



<a NAME="E1:23"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4: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> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4: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> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>
;<br/>
<a NAME="E3:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></span>;<br/>
<a NAME="E4:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E5:23"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4: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> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4: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> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>
;<br/>
<a NAME="E7:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="bcialg_1.html#T5" title="BCIALG_1:th.5">BCIALG_1:5</a></span>;<br/>
<a NAME="E8:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E9:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E10:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E11:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="bcialg_1.html#T3" title="BCIALG_1:th.3">BCIALG_1:3</a></span>;<br/>
<a NAME="E12:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E13:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E14:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E15:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>;<br/>
<a NAME="E16:23"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>;<br/>
<a NAME="E17:23"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>;<br/>
<a NAME="E18:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E19:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E20:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E21:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="bcialg_1.html#T3" title="BCIALG_1:th.3">BCIALG_1:3</a></span>;<br/>
<a NAME="E22:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<a NAME="E23:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E24:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_4.html#E8"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E25:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_4.html#T7" target="_self" title="BCIALG_4:th.7">Th7</a></span>;<br/>
<a NAME="E26:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></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_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
