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<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or ( not <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:32_1_1"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="arytm_1.html#T8" target="_self" title="ARYTM_1:th.8">Th8</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_2.html#T14" title="ARYTM_2:th.14">ARYTM_2:14</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="#E1:32_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E4:32_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="arytm_2.html#T12" title="ARYTM_2:th.12">ARYTM_2:12</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:32_1_2"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 <b>by </b><i><a class="ref" href="arytm_2.html#T4" title="ARYTM_2:th.4">ARYTM_2:4</a></i>;<br/><b>hence </b><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="#E1:32_1_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="arytm_2.html#T4" title="ARYTM_2:th.4">ARYTM_2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:32_1_2"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="arytm_1.html#E17"><i><font color="Green" title="E14">Lm4</font></i></a>, <a class="ref" href="arytm_2.html#T4" title="ARYTM_2:th.4">ARYTM_2:4</a></i>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_3"><b>suppose </b></a><a NAME="E1:32_1_3"/><i><font color="Green" title="E26">A4</font></i>: 
( not <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:32_1_3"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E11"><i><font color="Green" title="E9">Lm2</font></i></a></i>;<br/><a NAME="E3:32_1_3"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:32_1_3"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></i>;<br/><b>hence </b><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
<b>by </b><i><a class="ref" href="arytm_2.html#T4" title="ARYTM_2:th.4">ARYTM_2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:32_1_3"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></i>
;<br/> <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></div><b>end;</b></div>

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