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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Olive" title="b3">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b3">Z</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font>, <font color="Maroon" title="c6">Z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font></span>)</span> )</span><br/>

<a NAME="E4:48"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def19</a></span>;<br/>
<a NAME="E5:48"/>
( <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c6">Z</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def18</a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def19</a></span>; <a class="txt" 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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