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<b>let </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">x</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> ;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font> or ( not <font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">y</font> ) or ( not <font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font>
 ;<br/><div class="add"><a NAME="E2:40_1_1"/><b>then </b>
<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">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="arytm_1.html#E21"><i><font color="Green" title="E18">Lm8</font></i></a></i>;<br/><a NAME="E3:40_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="txt" href="arytm_1.html#E1:40_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="arytm_1.html#D2" target="_self" title="ARYTM_1:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E4:40_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E22"><i><font color="Green" title="E19">Lm9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:40_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
not <font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font>
 <b>and </b><br/><a NAME="E2:40_1_2"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">y</font>
 ;<br/><div class="add"><a NAME="E3:40_1_2"/><i><font color="Green" title="E30">A5</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">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="arytm_1.html#E1:40_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="arytm_1.html#E21"><i><font color="Green" title="E18">Lm8</font></i></a></i>;<br/><a NAME="E4:40_1_2"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="arytm_1.html#E2:40_1_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="arytm_1.html#T4" target="_self" title="ARYTM_1:th.4">Th4</a></i>;<br/><a NAME="E5:40_1_2"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">x</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="arytm_1.html#E13"><i><font color="Green" title="E11">Lm3</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">x</font> = 
<font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="arytm_1.html#E18"><i><font color="Green" title="E15">Lm5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_1.html#E5:40_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="arytm_1.html#E17"><i><font color="Green" title="E14">Lm4</font></i></a></i>
;<br/><b>hence </b><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_1.html#E1:40_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="arytm_1.html#E4:40_1_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="arytm_1.html#D2" target="_self" title="ARYTM_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="arytm_1.html#E3:40_1_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="arytm_1.html#D2" target="_self" title="ARYTM_1:def.2">Def2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:40_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
not <font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</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="c3">x</font>
 <b>and </b><br/><a NAME="E2:40_1_3"/><i><font color="Green" title="E29">A9</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">y</font>
 ;<br/><div class="add"><a NAME="E3:40_1_3"/><i><font color="Green" title="E30">A10</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">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="arytm_1.html#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="arytm_1.html#E1:40_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="arytm_1.html#E21"><i><font color="Green" title="E18">Lm8</font></i></a></i>;<br/><a NAME="E4:40_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_1.html#E2:40_1_3"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="arytm_1.html#E26"><i><font color="Green" title="E22">Lm11</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="arytm_1.html#E1:40_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="arytm_1.html#D2" target="_self" title="ARYTM_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="arytm_1.html#E3:40_1_3"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="arytm_1.html#D2" target="_self" title="ARYTM_1:def.2">Def2</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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