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<span class="kw">let </span><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_1"/>
( <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or ( not <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) or ( not <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:21_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></span>;<br/><a NAME="E3:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="arytm_1.html#T3" target="_self" title="ARYTM_1:th.3">Th3</a></span>;<br/><a NAME="E4:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="arytm_1.html#E21"><span class="lab"><font color="Green" title="E18">Lm8</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c2">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:21_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</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="21_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="arytm_1.html#E17"><span class="lab"><font color="Green" title="E14">Lm4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="arytm_2.html#D8" title="ARYTM_2:def.8">ARYTM_2:def 8</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="21_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_3"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
not <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_3"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><div class="add"><a NAME="E3:21_1_3"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></span>;<br/><a NAME="E4:21_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
not <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_3"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="arytm_1.html#T3" target="_self" title="ARYTM_1:th.3">Th3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_3_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_1_3_1"/>
<font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_3"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_3"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E1:21_1_3_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="arytm_1.html#T5" target="_self" title="ARYTM_1:th.5">Th5</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><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> = 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_3"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</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="21_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:21_1_4"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
not <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_4"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><div class="add"><a NAME="E3:21_1_4"/>
not <font color="Maroon" title="c4">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_4"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E2:21_1_4"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="arytm_1.html#E21"><span class="lab"><font color="Green" title="E18">Lm8</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <font color="Maroon" title="c4">x</font> = 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c2">z</font> <a href="arytm_1.html#K1" title="ARYTM_1:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_4"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="arytm_1.html#D1" target="_self" title="ARYTM_1:def.1">Def1</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>

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