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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E37">A1</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E2:49"/><b>then </b>
( <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a>, <a class="ref" href="mod_2.html#D19" target="_self" title="MOD_2:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E3:49"/>
( <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> )
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a>, <a class="ref" href="mod_2.html#D19" target="_self" title="MOD_2:def.19">Def19</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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