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<b>let </b><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">X3</font>, <font color="Maroon" title="c4">X4</font>, <font color="Maroon" title="c5">X5</font>, <font color="Maroon" title="c6">X6</font>, <font color="Maroon" title="c7">X7</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> ) implies <font color="Maroon" title="c1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> )
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:58"/>
( <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E4:58"/><b>then </b><i><font color="Green" title="E47">A3</font></i>: 
( <font color="Maroon" title="c1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">X3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">X4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">X5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">X6</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">X7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="mcart_3.html#T52" target="_self" title="MCART_3:th.52">MC352</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1"/>
( <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> or <font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_1_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font>
 <b>and </b><br/><a NAME="E4:58_1_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">x5</font>, <font color="Olive" title="b6">x6</font>, <font color="Olive" title="b7">x7</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> or <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> ) holds <br/><font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="mcart_3.html#K9" title="MCART_3:func.9">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font>,<font color="Olive" title="b5">x5</font>,<font color="Olive" title="b6">x6</font>,<font color="Olive" title="b7">x7</font></span><a href="mcart_3.html#K9" title="MCART_3:func.9">]</a></span>
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T46" target="_self" title="MCART_3:th.46">Th46</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <b>by </b><i><a class="txt" href="#E1:58_1_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="#E3:58_1_1_1"><i><font color="Green" title="E49">A5</font></i></a></i>;<br/><a NAME="E6:58_1_1_1"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_3.html#K9" title="MCART_3:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K11" title="MCART_3:func.11">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K12" title="MCART_3:func.12">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K13" title="MCART_3:func.13">`3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K14" title="MCART_3:func.14">`4</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K15" title="MCART_3:func.15">`5</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K16" title="MCART_3:func.16">`6</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K17" title="MCART_3:func.17">`7</a> </span>)</span></span><a href="mcart_3.html#K9" title="MCART_3:func.9">]</a></span> &amp; <font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K11" title="MCART_3:func.11">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> )
 <b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="mcart_3.html#T58" target="_self" title="MCART_3:th.58">Th58</a></i>;<br/><b>hence </b><a NAME="E7:58_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:58_1_1_1"><i><font color="Green" title="E50">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_2"/>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:58_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K2" title="MCART_3:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font></span><a href="mcart_3.html#K2" title="MCART_3:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_3.html#T12" target="_self" title="MCART_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T21" target="_self" title="MCART_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_3"><b>suppose </b></a><a NAME="E1:58_1_1_3"/>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:58_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K2" title="MCART_3:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="mcart_3.html#K2" title="MCART_3:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_3.html#T12" target="_self" title="MCART_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T21" target="_self" title="MCART_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_4"><b>suppose </b></a><a NAME="E1:58_1_1_4"/>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:58_1_1_4"/><b>then </b>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K2" title="MCART_3:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font></span><a href="mcart_3.html#K2" title="MCART_3:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_3.html#T12" target="_self" title="MCART_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T21" target="_self" title="MCART_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_5"><b>suppose </b></a><a NAME="E1:58_1_1_5"/>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:58_1_1_5"/><b>then </b>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K2" title="MCART_3:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font></span><a href="mcart_3.html#K2" title="MCART_3:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_3.html#T12" target="_self" title="MCART_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T21" target="_self" title="MCART_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6"><b>suppose </b></a><a NAME="E1:58_1_1_6"/>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:58_1_1_6"/><b>then </b>
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K2" title="MCART_3:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">X6</font>,<font color="Maroon" title="c7">X7</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="mcart_3.html#K2" title="MCART_3:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_3.html#T12" target="_self" title="MCART_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T21" target="_self" title="MCART_3:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_7"><b>suppose </b></a><a NAME="E1:58_1_1_7"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_1_1_7"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font>
 <b>and </b><br/><a NAME="E4:58_1_1_7"/><i><font color="Green" title="E50">A9</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">x5</font>, <font color="Olive" title="b6">x6</font>, <font color="Olive" title="b7">x7</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> or <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> ) holds <br/><font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="mcart_3.html#K9" title="MCART_3:func.9">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font>,<font color="Olive" title="b5">x5</font>,<font color="Olive" title="b6">x6</font>,<font color="Olive" title="b7">x7</font></span><a href="mcart_3.html#K9" title="MCART_3:func.9">]</a></span>
 <b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="mcart_3.html#T46" target="_self" title="MCART_3:th.46">Th46</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="mcart_3.html#K10" title="MCART_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">X7</font>,<font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font>,<font color="Maroon" title="c6">X6</font></span><a href="mcart_3.html#K10" title="MCART_3:func.10">:]</a></span> <b>by </b><i><a class="txt" href="#E1:58_1_1_7"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="#E3:58_1_1_7"><i><font color="Green" title="E49">A8</font></i></a></i>;<br/><a NAME="E6:58_1_1_7"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_3.html#K9" title="MCART_3:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K11" title="MCART_3:func.11">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K12" title="MCART_3:func.12">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K13" title="MCART_3:func.13">`3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K14" title="MCART_3:func.14">`4</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K15" title="MCART_3:func.15">`5</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K16" title="MCART_3:func.16">`6</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K17" title="MCART_3:func.17">`7</a> </span>)</span></span><a href="mcart_3.html#K9" title="MCART_3:func.9">]</a></span> &amp; <font color="Maroon" title="c9">y</font> <a href="mcart_3.html#K12" title="MCART_3:func.12">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> )
 <b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="mcart_3.html#T58" target="_self" title="MCART_3:th.58">Th58</a></i>;<br/><b>hence </b><a NAME="E7:58_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:58_1_1_7"><i><font color="Green" title="E50">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:58"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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