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<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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/>

<a NAME="E1:30"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#D1" title="NUMBERS:def.1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) or ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 ex <font color="Olive" title="b1">x99</font>, <font color="Olive" title="b2">y99</font> being   <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>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x99</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">y99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E5:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 ex <font color="Olive" title="b1">xy99</font>, <font color="Olive" title="b2">z99</font> being   <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>  st <br/>(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xy99</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xy99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">z99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E6:30_1_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 ex <font color="Olive" title="b1">y9</font>, <font color="Olive" title="b2">z9</font> being   <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>  st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y9</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">z9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E7:30_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">yz9</font> being   <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>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">yz9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:30_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E5:30_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_2"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E3:30_1_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 ex <font color="Olive" title="b1">x99</font>, <font color="Olive" title="b2">y99</font> being   <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>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x99</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">y99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">xy99</font>, <font color="Maroon" title="c8">z99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xy99</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_2"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">xy99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">y9</font>, <font color="Maroon" title="c10">z9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:30_1_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font>
 <span class="kw">and </span><br/><a NAME="E11:30_1_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">z9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:30_1_2"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E3:30_1_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E13:30_1_2"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c10">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E7:30_1_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_2_1_1"/>
( <font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font> or  not <font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E3:30_1_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">yz9</font> being   <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>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">yz9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_1_2_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E13:30_1_2"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="arytm_1.html#T20" title="ARYTM_1:th.20">ARYTM_1:20</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
not <font color="Maroon" title="c10">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E2:30_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A23</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">y9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E3:30_1_2_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yz9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="txt" href="#E6:30_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E13:30_1_2"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="arytm_1.html#T21" title="ARYTM_1:th.21">ARYTM_1:21</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E15:30_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_3"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_3"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:30_1_3"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">x99</font>, <font color="Maroon" title="c8">y99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_3"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x99</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_3"/><span class="lab"><font color="Green" title="E32">A30</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_3"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E2:30_1_3"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">z9</font>, <font color="Maroon" title="c10">y9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E9:30_1_3"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font>
 <span class="kw">and </span><br/><a NAME="E10:30_1_3"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:30_1_3"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_3"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E3:30_1_3"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E12:30_1_3"/><span class="lab"><font color="Green" title="E37">A35</font></span>: 
<font color="Maroon" title="c10">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_3"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E6:30_1_3"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_3_1_1"/>
( ( <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font> &amp; <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font> ) or ( <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font> &amp; not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font> ) or ( not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font> &amp; <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font> ) or ( not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font> &amp; not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E38">A36</font></span>: 
<font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E39">A37</font></span>: 
<font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E40">A38</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_3"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E41">A39</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E42">A40</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E7:30_1_3_1_1_1"/><span class="lab"><font color="Green" title="E43">A41</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_3"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E7:30_1_3"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E12:30_1_3"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E5:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E8:30_1_3_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z99</font>, <font color="Olive" title="b2">xy99</font> being   <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>  st <br/>( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">xy99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">xy99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:30_1_3_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_3"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E5:30_1_3"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="txt" href="#E3:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="txt" href="#E5:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="txt" href="#E6:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E42">A40</font></span></a>, <a class="txt" href="#E7:30_1_3_1_1_1"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="ref" href="arytm_1.html#T12" title="ARYTM_1:th.12">ARYTM_1:12</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E38">A42</font></span>: 
<font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E39">A43</font></span>: 
not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E40">A44</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_3"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E12:30_1_3"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A42</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E4:30_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A45</font></span>: 
 ex <font color="Olive" title="b1">z99</font>, <font color="Olive" title="b2">xy99</font> being   <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>  st <br/>( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">xy99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">xy99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E5:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E42">A46</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_3"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A43</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E6:30_1_3_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E43">A47</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">and </span><br/><a NAME="E9:30_1_3_1_1_2"/><span class="lab"><font color="Green" title="E44">A48</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yz9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E42">A46</font></span></a>, <a class="txt" href="#E9:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E44">A48</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_3"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E5:30_1_3"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A42</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A43</font></span></a>, <a class="txt" href="#E8:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E43">A47</font></span></a>, <a class="txt" href="#E3:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E40">A44</font></span></a>, <a class="txt" href="#E4:30_1_3_1_1_2"><span class="lab"><font color="Green" title="E41">A45</font></span></a>, <a class="ref" href="arytm_1.html#T22" title="ARYTM_1:th.22">ARYTM_1:22</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E38">A49</font></span>: 
not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E39">A50</font></span>: 
<font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E40">A51</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">x99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_3"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E12:30_1_3"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E4:30_1_3_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z99</font>, <font color="Maroon" title="c12">yx99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E41">A52</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E42">A53</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yx99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yx99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E8:30_1_3_1_1_3"/><span class="lab"><font color="Green" title="E43">A54</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_3"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E9:30_1_3_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">zy99</font> being   <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>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">zy99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">zy99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">hence </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">z99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">x99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_3"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E5:30_1_3"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E6:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="txt" href="#E8:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="ref" href="arytm_1.html#T22" title="ARYTM_1:th.22">ARYTM_1:22</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="txt" href="#E7:30_1_3_1_1_3"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
;<br/> <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E38">A55</font></span>: 
not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">x99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E39">A56</font></span>: 
not <font color="Maroon" title="c10">y9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E40">A57</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_3"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E4:30_1_3_1_1_4"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E41">A58</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E42">A59</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yz9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yz9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E8:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E43">A60</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">x99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_3"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E12:30_1_3"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E9:30_1_3_1_1_4"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z99</font>, <font color="Maroon" title="c14">yx99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E11:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E44">A61</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z99</font>
 <span class="kw">and </span><br/><a NAME="E12:30_1_3_1_1_4"/><span class="lab"><font color="Green" title="E45">A62</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">yx99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">yx99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c11">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E40">A57</font></span></a>, <a class="txt" href="#E7:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E42">A59</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<font color="Maroon" title="c13">z99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">x99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_3"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E5:30_1_3"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E1:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E38">A55</font></span></a>, <a class="txt" href="#E2:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E39">A56</font></span></a>, <a class="txt" href="#E11:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E44">A61</font></span></a>, <a class="txt" href="#E6:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E41">A58</font></span></a>, <a class="ref" href="arytm_1.html#T23" title="ARYTM_1:th.23">ARYTM_1:23</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E43">A60</font></span></a>, <a class="txt" href="#E12:30_1_3_1_1_4"><span class="lab"><font color="Green" title="E45">A62</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:30_1_3"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_4"/><span class="lab"><font color="Green" title="E28">A63</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_4"/><span class="lab"><font color="Green" title="E29">A64</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:30_1_4"/><span class="lab"><font color="Green" title="E30">A65</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_4"/>
( (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_4"><span class="lab"><font color="Green" title="E29">A64</font></span></a>, <a class="txt" href="#E3:30_1_4"><span class="lab"><font color="Green" title="E30">A65</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y9</font>, <font color="Maroon" title="c8">z9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_4"/><span class="lab"><font color="Green" title="E31">A66</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_4"/><span class="lab"><font color="Green" title="E32">A67</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">z9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_4"/><span class="lab"><font color="Green" title="E33">A68</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_4"><span class="lab"><font color="Green" title="E29">A64</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E9:30_1_4"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_4"><span class="lab"><font color="Green" title="E33">A68</font></span></a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x9</font>, <font color="Maroon" title="c10">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E11:30_1_4"/><span class="lab"><font color="Green" title="E34">A69</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">and </span><br/><a NAME="E12:30_1_4"/><span class="lab"><font color="Green" title="E35">A70</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">yz9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:30_1_4"/><span class="lab"><font color="Green" title="E36">A71</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">yz9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_4"><span class="lab"><font color="Green" title="E28">A63</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">x99</font>, <font color="Maroon" title="c12">y99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E15:30_1_4"/><span class="lab"><font color="Green" title="E37">A72</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x99</font>
 <span class="kw">and </span><br/><a NAME="E16:30_1_4"/><span class="lab"><font color="Green" title="E38">A73</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:30_1_4"/><span class="lab"><font color="Green" title="E39">A74</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_4"><span class="lab"><font color="Green" title="E28">A63</font></span></a>, <a class="txt" href="#E2:30_1_4"><span class="lab"><font color="Green" title="E29">A64</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E18:30_1_4"/><span class="lab"><font color="Green" title="E40">A75</font></span>: 
<font color="Maroon" title="c7">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_4"><span class="lab"><font color="Green" title="E31">A66</font></span></a>, <a class="txt" href="#E16:30_1_4"><span class="lab"><font color="Green" title="E38">A73</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_4_1_1"/>
( <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">x99</font> or  not <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">x99</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_4_1_1_1"/><span class="lab"><font color="Green" title="E41">A76</font></span>: 
<font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">x99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E2:30_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A77</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_4"><span class="lab"><font color="Green" title="E39">A74</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">xy99</font>, <font color="Maroon" title="c14">z99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:30_1_4_1_1_1"/><span class="lab"><font color="Green" title="E43">A78</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy99</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_4_1_1_1"/><span class="lab"><font color="Green" title="E44">A79</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:30_1_4_1_1_1"/><span class="lab"><font color="Green" title="E45">A80</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xy99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_4"><span class="lab"><font color="Green" title="E30">A65</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E7:30_1_4_1_1_1"/><span class="lab"><font color="Green" title="E46">A81</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_4"><span class="lab"><font color="Green" title="E32">A67</font></span></a>, <a class="txt" href="#E5:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E44">A79</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c9">x9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_4"><span class="lab"><font color="Green" title="E33">A68</font></span></a>, <a class="txt" href="#E12:30_1_4"><span class="lab"><font color="Green" title="E35">A70</font></span></a>, <a class="txt" href="#E13:30_1_4"><span class="lab"><font color="Green" title="E36">A71</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_4"><span class="lab"><font color="Green" title="E37">A72</font></span></a>, <a class="txt" href="#E11:30_1_4"><span class="lab"><font color="Green" title="E34">A69</font></span></a>, <a class="txt" href="#E18:30_1_4"><span class="lab"><font color="Green" title="E40">A75</font></span></a>, <a class="txt" href="#E1:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E41">A76</font></span></a>, <a class="txt" href="#E2:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E42">A77</font></span></a>, <a class="txt" href="#E4:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E43">A78</font></span></a>, <a class="txt" href="#E6:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E45">A80</font></span></a>, <a class="txt" href="#E7:30_1_4_1_1_1"><span class="lab"><font color="Green" title="E46">A81</font></span></a>, <a class="ref" href="arytm_1.html#T24" title="ARYTM_1:th.24">ARYTM_1:24</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E41">A82</font></span>: 
not <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">x99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E2:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E42">A83</font></span>: 
(  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_4"><span class="lab"><font color="Green" title="E30">A65</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E3:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E43">A84</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c12">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">x99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_4"><span class="lab"><font color="Green" title="E39">A74</font></span></a>, <a class="txt" href="#E1:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E4:30_1_4_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:30_1_4_1_1_2"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">xy99</font>, <font color="Maroon" title="c14">z99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E44">A85</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">xy99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E45">A86</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E46">A87</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">xy99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">z99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E43">A84</font></span></a>, <a class="txt" href="#E2:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E42">A83</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E47">A88</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_4"><span class="lab"><font color="Green" title="E32">A67</font></span></a>, <a class="txt" href="#E8:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E45">A86</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:30_1_4_1_1_2"/><span class="lab"><font color="Green" title="E48">A89</font></span>: 
<font color="Maroon" title="c10">yz9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_4"><span class="lab"><font color="Green" title="E33">A68</font></span></a>, <a class="txt" href="#E12:30_1_4"><span class="lab"><font color="Green" title="E35">A70</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E12:30_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">yz9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_4"><span class="lab"><font color="Green" title="E40">A75</font></span></a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></span>;<br/><a NAME="E13:30_1_4_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c10">yz9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_4"><span class="lab"><font color="Green" title="E37">A72</font></span></a>, <a class="txt" href="#E11:30_1_4"><span class="lab"><font color="Green" title="E34">A69</font></span></a>, <a class="txt" href="#E1:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></span>;<br/><span class="kw">hence </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y9</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_1_4"><span class="lab"><font color="Green" title="E36">A71</font></span></a>, <a class="txt" href="#E11:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E48">A89</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c14">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">x99</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_4"><span class="lab"><font color="Green" title="E37">A72</font></span></a>, <a class="txt" href="#E11:30_1_4"><span class="lab"><font color="Green" title="E34">A69</font></span></a>, <a class="txt" href="#E18:30_1_4"><span class="lab"><font color="Green" title="E40">A75</font></span></a>, <a class="txt" href="#E1:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E41">A82</font></span></a>, <a class="txt" href="#E10:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E47">A88</font></span></a>, <a class="ref" href="arytm_1.html#T13" title="ARYTM_1:th.13">ARYTM_1:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E43">A84</font></span></a>, <a class="txt" href="#E7:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E44">A85</font></span></a>, <a class="txt" href="#E9:30_1_4_1_1_2"><span class="lab"><font color="Green" title="E46">A87</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
;<br/> <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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E20:30_1_4"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_5"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_5"/><span class="lab"><font color="Green" title="E28">A90</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_5"/><span class="lab"><font color="Green" title="E29">A91</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E3:30_1_5"/><span class="lab"><font color="Green" title="E30">A92</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_5"/><span class="lab"><font color="Green" title="E31">A93</font></span>: 
 ex <font color="Olive" title="b1">z99</font>, <font color="Olive" title="b2">y99</font> being   <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>  st <br/>( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">y99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_5"><span class="lab"><font color="Green" title="E28">A90</font></span></a>, <a class="txt" href="#E2:30_1_5"><span class="lab"><font color="Green" title="E29">A91</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">zy99</font>, <font color="Maroon" title="c8">x99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_5"/><span class="lab"><font color="Green" title="E32">A94</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">zy99</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_5"/><span class="lab"><font color="Green" title="E33">A95</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_5"/><span class="lab"><font color="Green" title="E34">A96</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">zy99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_5"><span class="lab"><font color="Green" title="E30">A92</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">y9</font>, <font color="Maroon" title="c10">x9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:30_1_5"/><span class="lab"><font color="Green" title="E35">A97</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font>
 <span class="kw">and </span><br/><a NAME="E11:30_1_5"/><span class="lab"><font color="Green" title="E36">A98</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:30_1_5"/><span class="lab"><font color="Green" title="E37">A99</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_5"><span class="lab"><font color="Green" title="E29">A91</font></span></a>, <a class="txt" href="#E3:30_1_5"><span class="lab"><font color="Green" title="E30">A92</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E13:30_1_5"/><span class="lab"><font color="Green" title="E38">A100</font></span>: 
<font color="Maroon" title="c10">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_5"><span class="lab"><font color="Green" title="E36">A98</font></span></a>, <a class="txt" href="#E7:30_1_5"><span class="lab"><font color="Green" title="E33">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_5_1_1"/>
( <font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font> or  not <font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_5_1_1_1"/><span class="lab"><font color="Green" title="E39">A101</font></span>: 
<font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font></span><br/><div class="add"><a NAME="E2:30_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A102</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_5"><span class="lab"><font color="Green" title="E37">A99</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E3:30_1_5_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z9</font>, <font color="Olive" title="b2">yx9</font> being   <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>  st <br/>( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">yx9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Olive" title="b2">yx9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_5"><span class="lab"><font color="Green" title="E28">A90</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_1_5_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_5"><span class="lab"><font color="Green" title="E35">A97</font></span></a>, <a class="txt" href="#E4:30_1_5"><span class="lab"><font color="Green" title="E31">A93</font></span></a>, <a class="txt" href="#E6:30_1_5"><span class="lab"><font color="Green" title="E32">A94</font></span></a>, <a class="txt" href="#E8:30_1_5"><span class="lab"><font color="Green" title="E34">A96</font></span></a>, <a class="txt" href="#E13:30_1_5"><span class="lab"><font color="Green" title="E38">A100</font></span></a>, <a class="txt" href="#E1:30_1_5_1_1_1"><span class="lab"><font color="Green" title="E39">A101</font></span></a>, <a class="txt" href="#E2:30_1_5_1_1_1"><span class="lab"><font color="Green" title="E40">A102</font></span></a>, <a class="ref" href="arytm_1.html#T20" title="ARYTM_1:th.20">ARYTM_1:20</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_5_1_1_2"/><span class="lab"><font color="Green" title="E39">A103</font></span>: 
not <font color="Maroon" title="c10">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font></span><br/><div class="add"><a NAME="E2:30_1_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A104</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">y9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_5"><span class="lab"><font color="Green" title="E37">A99</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E3:30_1_5_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z9</font>, <font color="Maroon" title="c12">yx9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_5_1_1_2"/><span class="lab"><font color="Green" title="E41">A105</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_5_1_1_2"/><span class="lab"><font color="Green" title="E42">A106</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yx9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">yx9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_5"><span class="lab"><font color="Green" title="E28">A90</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<font color="Maroon" title="c11">z9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_5_1_1_2"><span class="lab"><font color="Green" title="E40">A104</font></span></a>, <a class="txt" href="#E6:30_1_5_1_1_2"><span class="lab"><font color="Green" title="E42">A106</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_5"><span class="lab"><font color="Green" title="E35">A97</font></span></a>, <a class="txt" href="#E4:30_1_5"><span class="lab"><font color="Green" title="E31">A93</font></span></a>, <a class="txt" href="#E6:30_1_5"><span class="lab"><font color="Green" title="E32">A94</font></span></a>, <a class="txt" href="#E8:30_1_5"><span class="lab"><font color="Green" title="E34">A96</font></span></a>, <a class="txt" href="#E13:30_1_5"><span class="lab"><font color="Green" title="E38">A100</font></span></a>, <a class="txt" href="#E1:30_1_5_1_1_2"><span class="lab"><font color="Green" title="E39">A103</font></span></a>, <a class="txt" href="#E5:30_1_5_1_1_2"><span class="lab"><font color="Green" title="E41">A105</font></span></a>, <a class="ref" href="arytm_1.html#T21" title="ARYTM_1:th.21">ARYTM_1:21</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E15:30_1_5"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_6"/><span class="lab"><font color="Green" title="E28">A107</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:30_1_6"/><span class="lab"><font color="Green" title="E29">A108</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E3:30_1_6"/><span class="lab"><font color="Green" title="E30">A109</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">y9</font>, <font color="Maroon" title="c8">z9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_6"/><span class="lab"><font color="Green" title="E31">A110</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_6"/><span class="lab"><font color="Green" title="E32">A111</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">z9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_6"/><span class="lab"><font color="Green" title="E33">A112</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c8">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_6"><span class="lab"><font color="Green" title="E29">A108</font></span></a>, <a class="txt" href="#E3:30_1_6"><span class="lab"><font color="Green" title="E30">A109</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">x99</font>, <font color="Maroon" title="c10">y99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E9:30_1_6"/><span class="lab"><font color="Green" title="E34">A113</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c9">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:30_1_6"/><span class="lab"><font color="Green" title="E35">A114</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y99</font>
 <span class="kw">and </span><br/><a NAME="E11:30_1_6"/><span class="lab"><font color="Green" title="E36">A115</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c9">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_6"><span class="lab"><font color="Green" title="E28">A107</font></span></a>, <a class="txt" href="#E2:30_1_6"><span class="lab"><font color="Green" title="E29">A108</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_6_1_1"/>
( ( <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font> &amp; <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font> ) or ( not <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font> &amp; <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font> ) or ( not <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font> &amp; <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font> ) or ( not <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font> &amp; not <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E37">A116</font></span>: 
<font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E38">A117</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E39">A118</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_6"><span class="lab"><font color="Green" title="E33">A112</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E38">A117</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E40">A119</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E41">A120</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz9</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yz9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c11">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_6"><span class="lab"><font color="Green" title="E28">A107</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E7:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E42">A121</font></span>: 
<font color="Maroon" title="c11">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_6"><span class="lab"><font color="Green" title="E34">A113</font></span></a>, <a class="txt" href="#E5:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E40">A119</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E8:30_1_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A122</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E11:30_1_6"><span class="lab"><font color="Green" title="E36">A115</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E37">A116</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z99</font>, <font color="Maroon" title="c14">xy99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E44">A123</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:30_1_6_1_1_1"/><span class="lab"><font color="Green" title="E45">A124</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy99</font> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xy99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">z99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_6"><span class="lab"><font color="Green" title="E30">A109</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E12:30_1_6_1_1_1"/>
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_6"><span class="lab"><font color="Green" title="E32">A111</font></span></a>, <a class="txt" href="#E10:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E44">A123</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_6_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E37">A116</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E38">A117</font></span></a>, <a class="txt" href="#E3:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E39">A118</font></span></a>, <a class="txt" href="#E6:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E41">A120</font></span></a>, <a class="txt" href="#E7:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E42">A121</font></span></a>, <a class="txt" href="#E8:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E43">A122</font></span></a>, <a class="txt" href="#E11:30_1_6_1_1_1"><span class="lab"><font color="Green" title="E45">A124</font></span></a>, <a class="ref" href="arytm_1.html#T25" title="ARYTM_1:th.25">ARYTM_1:25</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E37">A125</font></span>: 
not <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E38">A126</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E39">A127</font></span>: 
(  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_6"><span class="lab"><font color="Green" title="E30">A109</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E4:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E40">A128</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6"><span class="lab"><font color="Green" title="E36">A115</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E37">A125</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E5:30_1_6_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E6:30_1_6_1_1_2"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z99</font>, <font color="Maroon" title="c12">yx9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E41">A129</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E42">A130</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yx9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yx9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E40">A128</font></span></a>, <a class="txt" href="#E3:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E39">A127</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E43">A131</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_6"><span class="lab"><font color="Green" title="E32">A111</font></span></a>, <a class="txt" href="#E8:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E41">A129</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E44">A132</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_6"><span class="lab"><font color="Green" title="E33">A112</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E38">A126</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x9</font>, <font color="Maroon" title="c14">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E13:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E45">A133</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E46">A134</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yz9</font>
 <span class="kw">and </span><br/><a NAME="E15:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E47">A135</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yz9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_6"><span class="lab"><font color="Green" title="E28">A107</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E16:30_1_6_1_1_2"/><span class="lab"><font color="Green" title="E48">A136</font></span>: 
<font color="Maroon" title="c13">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_6"><span class="lab"><font color="Green" title="E34">A113</font></span></a>, <a class="txt" href="#E13:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E45">A133</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E17:30_1_6_1_1_2"/>
<font color="Maroon" title="c14">yz9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E44">A132</font></span></a>, <a class="txt" href="#E14:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E46">A134</font></span></a>, <a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></span>;<br/><a NAME="E18:30_1_6_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c13">x9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c14">yz9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E37">A125</font></span></a>, <a class="txt" href="#E16:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E48">A136</font></span></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></span>;<br/><span class="kw">hence </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">x9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c8">z9</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E44">A132</font></span></a>, <a class="txt" href="#E14:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E46">A134</font></span></a>, <a class="txt" href="#E15:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E47">A135</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y99</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">z99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E37">A125</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E38">A126</font></span></a>, <a class="txt" href="#E16:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E48">A136</font></span></a>, <a class="txt" href="#E10:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E43">A131</font></span></a>, <a class="ref" href="arytm_1.html#T14" title="ARYTM_1:th.14">ARYTM_1:14</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E40">A128</font></span></a>, <a class="txt" href="#E9:30_1_6_1_1_2"><span class="lab"><font color="Green" title="E42">A130</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
;<br/> <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E37">A137</font></span>: 
not <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E38">A138</font></span>: 
<font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E39">A139</font></span>: 
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_6"><span class="lab"><font color="Green" title="E28">A107</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E4:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E40">A140</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_6"><span class="lab"><font color="Green" title="E33">A112</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E37">A137</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E5:30_1_6_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E6:30_1_6_1_1_3"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font>, <font color="Maroon" title="c12">yz99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E8:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E41">A141</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E42">A142</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yz99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yz99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E40">A140</font></span></a>, <a class="txt" href="#E3:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E39">A139</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E43">A143</font></span>: 
<font color="Maroon" title="c9">x99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_6"><span class="lab"><font color="Green" title="E34">A113</font></span></a>, <a class="txt" href="#E8:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E41">A141</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E44">A144</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6"><span class="lab"><font color="Green" title="E36">A115</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E38">A138</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z99</font>, <font color="Maroon" title="c14">yx99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E13:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E45">A145</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E46">A146</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yx99</font>
 <span class="kw">and </span><br/><a NAME="E15:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E47">A147</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">yx99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c13">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_6"><span class="lab"><font color="Green" title="E30">A109</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E16:30_1_6_1_1_3"/><span class="lab"><font color="Green" title="E48">A148</font></span>: 
<font color="Maroon" title="c13">z99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_6"><span class="lab"><font color="Green" title="E32">A111</font></span></a>, <a class="txt" href="#E13:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E45">A145</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E17:30_1_6_1_1_3"/>
<font color="Maroon" title="c14">yx99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E44">A144</font></span></a>, <a class="txt" href="#E14:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E46">A146</font></span></a>, <a class="ref" href="arytm_1.html#T11" title="ARYTM_1:th.11">ARYTM_1:11</a></span>;<br/><a NAME="E18:30_1_6_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A149</font></span>: 
not <font color="Maroon" title="c13">z99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c14">yx99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E37">A137</font></span></a>, <a class="txt" href="#E16:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E48">A148</font></span></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y9</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">x9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E40">A140</font></span></a>, <a class="txt" href="#E9:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E42">A142</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">z99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">x99</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E37">A137</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E38">A138</font></span></a>, <a class="txt" href="#E16:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E48">A148</font></span></a>, <a class="txt" href="#E10:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E43">A143</font></span></a>, <a class="ref" href="arytm_1.html#T14" title="ARYTM_1:th.14">ARYTM_1:14</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E44">A144</font></span></a>, <a class="txt" href="#E14:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E46">A146</font></span></a>, <a class="txt" href="#E15:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E47">A147</font></span></a>, <a class="txt" href="#E18:30_1_6_1_1_3"><span class="lab"><font color="Green" title="E49">A149</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_6_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E37">A150</font></span>: 
not <font color="Maroon" title="c9">x99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">y99</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E38">A151</font></span>: 
not <font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c7">y9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E3:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E39">A152</font></span>: 
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_6"><span class="lab"><font color="Green" title="E28">A107</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E4:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E40">A153</font></span>: 
(  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_6"><span class="lab"><font color="Green" title="E30">A109</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E41">A154</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_6"><span class="lab"><font color="Green" title="E36">A115</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E37">A150</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E6:30_1_6_1_1_4"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E7:30_1_6_1_1_4"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z99</font>, <font color="Maroon" title="c12">yx9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E9:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E42">A155</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E43">A156</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">yx9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">yx9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E41">A154</font></span></a>, <a class="txt" href="#E4:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E40">A153</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E44">A157</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_6"><span class="lab"><font color="Green" title="E32">A111</font></span></a>, <a class="txt" href="#E9:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E42">A155</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E12:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E45">A158</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_6"><span class="lab"><font color="Green" title="E33">A112</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E38">A151</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E13:30_1_6_1_1_4"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E14:30_1_6_1_1_4"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x9</font>, <font color="Maroon" title="c14">yz99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E16:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E46">A159</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E47">A160</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">yz99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">yz99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E45">A158</font></span></a>, <a class="txt" href="#E3:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E39">A152</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E18:30_1_6_1_1_4"/><span class="lab"><font color="Green" title="E48">A161</font></span>: 
<font color="Maroon" title="c13">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_6"><span class="lab"><font color="Green" title="E34">A113</font></span></a>, <a class="txt" href="#E16:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E46">A159</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c7">y9</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c13">x9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E45">A158</font></span></a>, <a class="txt" href="#E17:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E47">A160</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c10">y99</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">z99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6"><span class="lab"><font color="Green" title="E31">A110</font></span></a>, <a class="txt" href="#E10:30_1_6"><span class="lab"><font color="Green" title="E35">A114</font></span></a>, <a class="txt" href="#E1:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E37">A150</font></span></a>, <a class="txt" href="#E2:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E38">A151</font></span></a>, <a class="txt" href="#E11:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E44">A157</font></span></a>, <a class="txt" href="#E18:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E48">A161</font></span></a>, <a class="ref" href="arytm_1.html#T15" title="ARYTM_1:th.15">ARYTM_1:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E41">A154</font></span></a>, <a class="txt" href="#E10:30_1_6_1_1_4"><span class="lab"><font color="Green" title="E43">A156</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:30_1_6"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_7"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_7"/><span class="lab"><font color="Green" title="E28">A162</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_7"/><span class="lab"><font color="Green" title="E29">A163</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:30_1_7"/><span class="lab"><font color="Green" title="E30">A164</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_7"/>
( (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_7"><span class="lab"><font color="Green" title="E29">A163</font></span></a>, <a class="txt" href="#E3:30_1_7"><span class="lab"><font color="Green" title="E30">A164</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y9</font>, <font color="Maroon" title="c8">x9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E6:30_1_7"/><span class="lab"><font color="Green" title="E31">A165</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_7"/><span class="lab"><font color="Green" title="E32">A166</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_7"/><span class="lab"><font color="Green" title="E33">A167</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">x9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_7"><span class="lab"><font color="Green" title="E29">A163</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E9:30_1_7"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_7"><span class="lab"><font color="Green" title="E33">A167</font></span></a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">z9</font>, <font color="Maroon" title="c10">yx9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E11:30_1_7"/><span class="lab"><font color="Green" title="E34">A168</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font>
 <span class="kw">and </span><br/><a NAME="E12:30_1_7"/><span class="lab"><font color="Green" title="E35">A169</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">yx9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:30_1_7"/><span class="lab"><font color="Green" title="E36">A170</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c10">yx9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_7"><span class="lab"><font color="Green" title="E28">A162</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">z99</font>, <font color="Maroon" title="c12">y99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E15:30_1_7"/><span class="lab"><font color="Green" title="E37">A171</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font>
 <span class="kw">and </span><br/><a NAME="E16:30_1_7"/><span class="lab"><font color="Green" title="E38">A172</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:30_1_7"/><span class="lab"><font color="Green" title="E39">A173</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_7"><span class="lab"><font color="Green" title="E28">A162</font></span></a>, <a class="txt" href="#E2:30_1_7"><span class="lab"><font color="Green" title="E29">A163</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E18:30_1_7"/><span class="lab"><font color="Green" title="E40">A174</font></span>: 
<font color="Maroon" title="c7">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_7"><span class="lab"><font color="Green" title="E31">A165</font></span></a>, <a class="txt" href="#E16:30_1_7"><span class="lab"><font color="Green" title="E38">A172</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_7_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_7_1_1"/>
( <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z99</font> or  not <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z99</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_7_1_1_1"/><span class="lab"><font color="Green" title="E41">A175</font></span>: 
<font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font></span><br/><div class="add"><a NAME="E2:30_1_7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A176</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_7"><span class="lab"><font color="Green" title="E39">A173</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">zy99</font>, <font color="Maroon" title="c14">x99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:30_1_7_1_1_1"/><span class="lab"><font color="Green" title="E43">A177</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zy99</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_7_1_1_1"/><span class="lab"><font color="Green" title="E44">A178</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:30_1_7_1_1_1"/><span class="lab"><font color="Green" title="E45">A179</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">zy99</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <font color="Maroon" title="c14">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_7"><span class="lab"><font color="Green" title="E30">A164</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E7:30_1_7_1_1_1"/><span class="lab"><font color="Green" title="E46">A180</font></span>: 
<font color="Maroon" title="c8">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_7"><span class="lab"><font color="Green" title="E32">A166</font></span></a>, <a class="txt" href="#E5:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E44">A178</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<font color="Maroon" title="c9">z9</font> <a href="arytm_1.html#K2" title="ARYTM_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_7"><span class="lab"><font color="Green" title="E33">A167</font></span></a>, <a class="txt" href="#E12:30_1_7"><span class="lab"><font color="Green" title="E35">A169</font></span></a>, <a class="txt" href="#E13:30_1_7"><span class="lab"><font color="Green" title="E36">A170</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_7"><span class="lab"><font color="Green" title="E37">A171</font></span></a>, <a class="txt" href="#E11:30_1_7"><span class="lab"><font color="Green" title="E34">A168</font></span></a>, <a class="txt" href="#E18:30_1_7"><span class="lab"><font color="Green" title="E40">A174</font></span></a>, <a class="txt" href="#E1:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E41">A175</font></span></a>, <a class="txt" href="#E2:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E42">A176</font></span></a>, <a class="txt" href="#E4:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E43">A177</font></span></a>, <a class="txt" href="#E6:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E45">A179</font></span></a>, <a class="txt" href="#E7:30_1_7_1_1_1"><span class="lab"><font color="Green" title="E46">A180</font></span></a>, <a class="ref" href="arytm_1.html#T24" title="ARYTM_1:th.24">ARYTM_1:24</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E41">A181</font></span>: 
not <font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c11">z99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font></span><br/><div class="add"><a NAME="E2:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E42">A182</font></span>: 
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_7"><span class="lab"><font color="Green" title="E30">A164</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E3:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E43">A183</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c12">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">z99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_7"><span class="lab"><font color="Green" title="E39">A173</font></span></a>, <a class="txt" href="#E1:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E41">A181</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>;<br/><a NAME="E4:30_1_7_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E5:30_1_7_1_1_2"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">zy99</font>, <font color="Maroon" title="c14">x99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E44">A184</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">zy99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E45">A185</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E46">A186</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">zy99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">x99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E43">A183</font></span></a>, <a class="txt" href="#E2:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E42">A182</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E47">A187</font></span>: 
<font color="Maroon" title="c8">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_7"><span class="lab"><font color="Green" title="E32">A166</font></span></a>, <a class="txt" href="#E8:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E45">A185</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:30_1_7_1_1_2"/><span class="lab"><font color="Green" title="E48">A188</font></span>: 
<font color="Maroon" title="c10">yx9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_7"><span class="lab"><font color="Green" title="E33">A167</font></span></a>, <a class="txt" href="#E12:30_1_7"><span class="lab"><font color="Green" title="E35">A169</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E12:30_1_7_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c10">yx9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_7"><span class="lab"><font color="Green" title="E40">A174</font></span></a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></span>;<br/><a NAME="E13:30_1_7_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c10">yx9</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c9">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_7"><span class="lab"><font color="Green" title="E37">A171</font></span></a>, <a class="txt" href="#E11:30_1_7"><span class="lab"><font color="Green" title="E34">A168</font></span></a>, <a class="txt" href="#E1:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E41">A181</font></span></a>, <a class="ref" href="arytm_1.html#T3" title="ARYTM_1:th.3">ARYTM_1:3</a></span>;<br/><span class="kw">hence </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y9</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c9">z9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_1_7"><span class="lab"><font color="Green" title="E36">A170</font></span></a>, <a class="txt" href="#E11:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E48">A188</font></span></a>, <a class="ref" href="arytm_1.html#D2" title="ARYTM_1:def.2">ARYTM_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">y99</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c11">z99</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_1_7"><span class="lab"><font color="Green" title="E37">A171</font></span></a>, <a class="txt" href="#E11:30_1_7"><span class="lab"><font color="Green" title="E34">A168</font></span></a>, <a class="txt" href="#E18:30_1_7"><span class="lab"><font color="Green" title="E40">A174</font></span></a>, <a class="txt" href="#E1:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E41">A181</font></span></a>, <a class="txt" href="#E10:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E47">A187</font></span></a>, <a class="ref" href="arytm_1.html#T13" title="ARYTM_1:th.13">ARYTM_1:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E43">A183</font></span></a>, <a class="txt" href="#E7:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E44">A184</font></span></a>, <a class="txt" href="#E9:30_1_7_1_1_2"><span class="lab"><font color="Green" title="E46">A186</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
;<br/> <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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E20:30_1_7"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_8"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:30_1_8"/><span class="lab"><font color="Green" title="E28">A189</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:30_1_8"/><span class="lab"><font color="Green" title="E29">A190</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:30_1_8"/><span class="lab"><font color="Green" title="E30">A191</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font></span><br/><div class="add"><a NAME="E4:30_1_8"/><span class="lab"><font color="Green" title="E31">A192</font></span>: 
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_8"><span class="lab"><font color="Green" title="E28">A189</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:30_1_8"/>
( (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_8"><span class="lab"><font color="Green" title="E29">A190</font></span></a>, <a class="txt" href="#E3:30_1_8"><span class="lab"><font color="Green" title="E30">A191</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y9</font>, <font color="Maroon" title="c8">z9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:30_1_8"/><span class="lab"><font color="Green" title="E32">A193</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1_8"/><span class="lab"><font color="Green" title="E33">A194</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">z9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_8"/><span class="lab"><font color="Green" title="E34">A195</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c8">z9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_8"><span class="lab"><font color="Green" title="E29">A190</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E10:30_1_8"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_8"><span class="lab"><font color="Green" title="E34">A195</font></span></a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:30_1_8"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x9</font>, <font color="Maroon" title="c10">yz9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E13:30_1_8"/><span class="lab"><font color="Green" title="E35">A196</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c9">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:30_1_8"/><span class="lab"><font color="Green" title="E36">A197</font></span>: 
(  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">yz9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c10">yz9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_8"><span class="lab"><font color="Green" title="E34">A195</font></span></a>, <a class="txt" href="#E4:30_1_8"><span class="lab"><font color="Green" title="E31">A192</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E15:30_1_8"/><span class="lab"><font color="Green" title="E37">A198</font></span>: 
(  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_8"><span class="lab"><font color="Green" title="E30">A191</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E16:30_1_8"/>
( (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_8"><span class="lab"><font color="Green" title="E28">A189</font></span></a>, <a class="txt" href="#E2:30_1_8"><span class="lab"><font color="Green" title="E29">A190</font></span></a>, <a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x99</font>, <font color="Maroon" title="c12">y99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E18:30_1_8"/><span class="lab"><font color="Green" title="E38">A199</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">x99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E19:30_1_8"/><span class="lab"><font color="Green" title="E39">A200</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E20:30_1_8"/><span class="lab"><font color="Green" title="E40">A201</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c12">y99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_8"><span class="lab"><font color="Green" title="E28">A189</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a></span>;<br/><a NAME="E21:30_1_8"/><span class="lab"><font color="Green" title="E41">A202</font></span>: 
<font color="Maroon" title="c9">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:30_1_8"><span class="lab"><font color="Green" title="E38">A199</font></span></a>, <a class="txt" href="#E13:30_1_8"><span class="lab"><font color="Green" title="E35">A196</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E22:30_1_8"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_1_8"><span class="lab"><font color="Green" title="E40">A201</font></span></a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E23:30_1_8"/><span class="kw">then </span>
(  not  <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  or  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T5" target="_self" title="ARYTM_0:th.5">Th5</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">xy99</font>, <font color="Maroon" title="c14">z99</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E25:30_1_8"/><span class="lab"><font color="Green" title="E42">A203</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c13">xy99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E26:30_1_8"/><span class="lab"><font color="Green" title="E43">A204</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">z99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E27:30_1_8"/><span class="lab"><font color="Green" title="E44">A205</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c13">xy99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c14">z99</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_1_8"><span class="lab"><font color="Green" title="E40">A201</font></span></a>, <a class="txt" href="#E15:30_1_8"><span class="lab"><font color="Green" title="E37">A198</font></span></a>, <a class="ref" href="arytm_0.html#D2" target="_self" title="ARYTM_0:def.2">Def2</a>, <a class="txt" href="arytm_0.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E28:30_1_8"/><span class="lab"><font color="Green" title="E45">A206</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_8"><span class="lab"><font color="Green" title="E33">A194</font></span></a>, <a class="txt" href="#E26:30_1_8"><span class="lab"><font color="Green" title="E43">A204</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E29:30_1_8"/><span class="lab"><font color="Green" title="E46">A207</font></span>: 
<font color="Maroon" title="c7">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_8"><span class="lab"><font color="Green" title="E32">A193</font></span></a>, <a class="txt" href="#E19:30_1_8"><span class="lab"><font color="Green" title="E39">A200</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">thus </span> <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">z9</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c7">y9</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c9">x9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_8"><span class="lab"><font color="Green" title="E34">A195</font></span></a>, <a class="txt" href="#E14:30_1_8"><span class="lab"><font color="Green" title="E36">A197</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c14">z99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">y99</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c11">x99</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_8"><span class="lab"><font color="Green" title="E45">A206</font></span></a>, <a class="txt" href="#E21:30_1_8"><span class="lab"><font color="Green" title="E41">A202</font></span></a>, <a class="txt" href="#E29:30_1_8"><span class="lab"><font color="Green" title="E46">A207</font></span></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></span>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30_1_8"><span class="lab"><font color="Green" title="E40">A201</font></span></a>, <a class="txt" href="#E25:30_1_8"><span class="lab"><font color="Green" title="E42">A203</font></span></a>, <a class="txt" href="#E27:30_1_8"><span class="lab"><font color="Green" title="E44">A205</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
