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<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font>, <font color="Maroon" title="c4">P19</font>, <font color="Maroon" title="c5">P29</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">P19</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c5">P29</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c4">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">P29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) holds <br/>( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">P19</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c5">P29</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c4">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">P29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) implies ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> 
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>
 <span class="kw">and </span><br/><a NAME="E3:64"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>
 <span class="kw">and </span><br/><a NAME="E4:64"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E5:64"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c4">P19</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>
 <span class="kw">and </span><br/><a NAME="E6:64"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c5">P29</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>
 <span class="kw">and </span><br/><a NAME="E7:64"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c4">P19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">P29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">P</font> = <font color="Maroon" title="c1">P</font> as   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<a NAME="E9:64"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="jordan6.html#T49" target="_self" title="JORDAN6:th.49">Th49</a></span>;<br/>
<a NAME="E10:64"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E11:64"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E12:64"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E13:64"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E14:64"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E15:64"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E16:64"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E17:64"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<a NAME="E18:64"/>
<font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Q1</font> = <font color="Maroon" title="c2">P1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:64"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E17:64"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:64"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E21:64"/>
<font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Q2</font> = <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:64"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E20:64"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E23:64"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c4">P19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E24:64"/>
<font color="Maroon" title="c4">P19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Q19</font> = <font color="Maroon" title="c4">P19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:64"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E23:64"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E26:64"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon" title="c5">P29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E27:64"/>
<font color="Maroon" title="c5">P29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Q29</font> = <font color="Maroon" title="c5">P29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:64"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E26:64"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E29:64"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></span>;<br/>
<a NAME="E30:64"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></span>;<br/>
<a NAME="E31:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A22</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E32:64"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E33:64"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E34:64"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E35:64"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">R1</font> = <font color="Maroon" title="c9">Q1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E37:64"/>
<font color="Maroon" title="c13">R1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="jordan6.html#T53" target="_self" title="JORDAN6:th.53">Th53</a></span>;<br/>
<a NAME="E38:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<font color="Maroon" title="c9">Q1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="jordan6.html#T54" target="_self" title="JORDAN6:th.54">Th54</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E39:64"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">R2</font> = <font color="Maroon" title="c10">Q2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E41:64"/>
<font color="Maroon" title="c14">R2</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="jordan6.html#T53" target="_self" title="JORDAN6:th.53">Th53</a></span>;<br/>
<a NAME="E42:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A27</font></span>: 
<font color="Maroon" title="c10">Q2</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="jordan6.html#T54" target="_self" title="JORDAN6:th.54">Th54</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E43:64"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">R19</font> = <font color="Maroon" title="c11">Q19</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P19</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E45:64"/>
<font color="Maroon" title="c15">R19</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="jordan6.html#T53" target="_self" title="JORDAN6:th.53">Th53</a></span>;<br/>
<a NAME="E46:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A28</font></span>: 
<font color="Maroon" title="c11">Q19</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="jordan6.html#T54" target="_self" title="JORDAN6:th.54">Th54</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E47:64"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">R29</font> = <font color="Maroon" title="c12">Q29</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P29</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E49:64"/>
<font color="Maroon" title="c16">R29</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="jordan6.html#T53" target="_self" title="JORDAN6:th.53">Th53</a></span>;<br/>
<a NAME="E50:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A29</font></span>: 
<font color="Maroon" title="c12">Q29</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="jordan6.html#T54" target="_self" title="JORDAN6:th.54">Th54</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E51:64"/><span class="lab"><font color="Green" title="E62">A30</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI: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"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font></span><br/>

<span class="kw">hence </span><a NAME="E2:64_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E15:64"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI: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>
<a NAME="E52:64"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI: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"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font></span><br/>

<span class="kw">hence </span><a NAME="E2:64_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:64"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E13:64"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI: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>
<a NAME="E53:64"/><span class="lab"><font color="Green" title="E64">A32</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P19</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P19</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI: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"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P19</font></span><br/>

<span class="kw">hence </span><a NAME="E2:64_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E11:64"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI: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>
<a NAME="E54:64"/><span class="lab"><font color="Green" title="E65">A33</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P29</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P29</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_4"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI: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"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P29</font></span><br/>

<a NAME="E2:64_4"/><span class="kw">then </span>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:64_4"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</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>
<a NAME="E55:64"/><span class="lab"><font color="Green" title="E66">A34</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:64"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E56:64"/><span class="lab"><font color="Green" title="E67">A35</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:64"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E57:64"/><span class="lab"><font color="Green" title="E68">A36</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:64"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E58:64"/><span class="lab"><font color="Green" title="E69">A37</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:64"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_5"/><span class="lab"><font color="Green" title="E70">A38</font></span>: 
(  not <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> or  not <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_5_1_1"/>
( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P29</font> or <font color="Maroon" title="c3">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5"><span class="lab"><font color="Green" title="E70">A38</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_5_1_1_1"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_1"/><span class="lab"><font color="Green" title="E71">A39</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E72">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_5_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">Q29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:64_5_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E3:64_5_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_1_1"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E4:64_5_1_1_1_1"/>
<font color="Maroon" title="c12">Q29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_5_1_1_1_1"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_5_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:64"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="txt" href="#E58:64"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E1:64_5_1_1_1"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E3:64_5_1_1_1_1"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="hs2(this)" href="javascript:()" title="64_5_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_5_1_1_1_2_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="E1:64_5_1_1_1_2_1"/>
( <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">Q29</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>  or <font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">Q1</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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_5_1_1_1"><span class="lab"><font color="Green" title="E72">A40</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_5_1_1_1_2_1_1"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E73">A44</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">Q29</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">Q29</font>;<br/><a NAME="E3:64_5_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E74">A45</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A44</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:64_5_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A46</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:64_5_1_1_1_2_1_1"/>
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E73">A44</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:64_5_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E75">A46</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:64_5_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">Q19</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E74">A45</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:64_5_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A47</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">Q2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">Q19</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="E2:64_5_1_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q19</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:64"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E42:64"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="#E46:64"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E8:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E76">A47</font></span></a>, <a class="ref" href="jordan1.html#T7" title="JORDAN1:th.7">JORDAN1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_5_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E14:64"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E15:64"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E16:64"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E31:64"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="jordan6.html#T60" target="_self" title="JORDAN6:th.60">Th60</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</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><a NAME="E10:64_5_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A48</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">Q19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E11:64_5_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E78">A49</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q29</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E79">A50</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font></span><br/>

<a NAME="E2:64_5_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E1:64_5_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A50</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E12:64_5_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">Q19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E79">A51</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font></span><br/>

<a NAME="E2:64_5_1_1_1_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_1_2_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_1_2_1_1_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="txt" href="#E1:64_5_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E79">A51</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E13:64_5_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E55:64"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="txt" href="#E56:64"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E57:64"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E58:64"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E11:64_5_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E78">A49</font></span></a>, <a class="ref" href="jordan6.html#T59" target="_self" title="JORDAN6:th.59">Th59</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</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="64_5_1_1_1_2_1_2"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E73">A52</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">Q1</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">Q1</font>;<br/><a NAME="E3:64_5_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E74">A53</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A52</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:64_5_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A54</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:64_5_1_1_1_2_1_2"/>
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E73">A52</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:64_5_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:64_5_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">Q2</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:64_5_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A55</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c11">Q19</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">Q2</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="E2:64_5_1_1_1_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q19</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:64"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="#E46:64"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E50:64"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="txt" href="#E8:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E76">A55</font></span></a>, <a class="ref" href="jordan1.html#T7" title="JORDAN1:th.7">JORDAN1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_5_1_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E14:64"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E15:64"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E16:64"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="txt" href="#E34:64"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="ref" href="jordan6.html#T60" target="_self" title="JORDAN6:th.60">Th60</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</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><a NAME="E10:64_5_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A56</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E11:64_5_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E78">A57</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_2_2"/><span class="lab"><font color="Green" title="E79">A58</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font></span><br/>

<a NAME="E2:64_5_1_1_1_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_1_2_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_1_2_1_2_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E1:64_5_1_1_1_2_1_2_2"><span class="lab"><font color="Green" title="E79">A58</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E12:64_5_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Q29</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_1_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_1_2_1_2_3"/><span class="lab"><font color="Green" title="E79">A59</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font></span><br/>

<a NAME="E2:64_5_1_1_1_2_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_1_2_1_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_1_2_1_2_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E1:64_5_1_1_1_2_1_2_3"><span class="lab"><font color="Green" title="E79">A59</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E13:64_5_1_1_1_2_1_2"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E55:64"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="txt" href="#E56:64"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E57:64"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E58:64"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E11:64_5_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="ref" href="jordan6.html#T59" target="_self" title="JORDAN6:th.59">Th59</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</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="E4:64_5_1_1_1"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</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="64_5_1_1_2"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_2"/><span class="lab"><font color="Green" title="E71">A60</font></span>: 
<font color="Maroon" title="c3">P2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">P19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E72">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_5_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A62</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">Q19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:64_5_1_1_2_1"/><span class="lab"><font color="Green" title="E74">A63</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E3:64_5_1_1_2_1"/><span class="lab"><font color="Green" title="E75">A64</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_2_1"><span class="lab"><font color="Green" title="E73">A62</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E4:64_5_1_1_2_1"/>
<font color="Maroon" title="c11">Q19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_5_1_1_2_1"><span class="lab"><font color="Green" title="E74">A63</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_5_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:64"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E57:64"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E1:64_5_1_1_2"><span class="lab"><font color="Green" title="E71">A60</font></span></a>, <a class="txt" href="#E3:64_5_1_1_2_1"><span class="lab"><font color="Green" title="E75">A64</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="hs2(this)" href="javascript:()" title="64_5_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_5_1_1_2_2_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="E1:64_5_1_1_2_2_1"/>
( <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">Q19</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>  or <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Q2</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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_5_1_1_2"><span class="lab"><font color="Green" title="E72">A61</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_5_1_1_2_2_1_1"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">Q19</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">Q19</font>;<br/><a NAME="E3:64_5_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E74">A66</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E73">A65</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:64_5_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A67</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:64_5_1_1_2_2_1_1"/>
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E73">A65</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:64_5_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:64_5_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Q29</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:64_5_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A68</font></span>: 
<font color="Maroon" title="c10">Q2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c9">Q1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">Q29</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="E2:64_5_1_1_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:64"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E42:64"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="#E50:64"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="txt" href="#E8:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E76">A68</font></span></a>, <a class="ref" href="jordan1.html#T7" title="JORDAN1:th.7">JORDAN1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_5_1_1_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E14:64"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E15:64"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E16:64"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E32:64"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="jordan6.html#T60" target="_self" title="JORDAN6:th.60">Th60</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</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><a NAME="E10:64_5_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A69</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Q29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E11:64_5_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E78">A70</font></span>: 
<font color="Maroon" title="c9">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Q19</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E79">A71</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font></span><br/>

<a NAME="E2:64_5_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_2_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E77">A69</font></span></a>, <a class="txt" href="#E1:64_5_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E79">A71</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E12:64_5_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">Q29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E79">A72</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font></span><br/>

<a NAME="E2:64_5_1_1_2_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_2_2_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_2_2_1_1_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E77">A69</font></span></a>, <a class="txt" href="#E1:64_5_1_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E79">A72</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E13:64_5_1_1_2_2_1_1"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E55:64"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="txt" href="#E56:64"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E57:64"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E58:64"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E11:64_5_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E78">A70</font></span></a>, <a class="ref" href="jordan6.html#T59" target="_self" title="JORDAN6:th.59">Th59</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</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="64_5_1_1_2_2_1_2"><span class="kw">case </span></a><a NAME="E1:64_5_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E73">A73</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Q2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Q2</font>;<br/><a NAME="E3:64_5_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E74">A74</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A73</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:64_5_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A75</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:64_5_1_1_2_2_1_2"/>
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A73</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:64_5_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E75">A75</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:64_5_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">Q1</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E74">A74</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:64_5_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A76</font></span>: 
<font color="Maroon" title="c11">Q19</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c12">Q29</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">Q1</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="E2:64_5_1_1_2_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:64"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E46:64"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E50:64"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="txt" href="#E8:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E76">A76</font></span></a>, <a class="ref" href="jordan1.html#T7" title="JORDAN1:th.7">JORDAN1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_5_1_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E14:64"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E15:64"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E16:64"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="txt" href="#E33:64"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="jordan6.html#T60" target="_self" title="JORDAN6:th.60">Th60</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</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><a NAME="E10:64_5_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A77</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E11:64_5_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E78">A78</font></span>: 
<font color="Maroon" title="c12">Q29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_2_2"/><span class="lab"><font color="Green" title="E79">A79</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font></span><br/>

<a NAME="E2:64_5_1_1_2_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">Q19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_2_2_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_2_2_1_2_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E77">A77</font></span></a>, <a class="txt" href="#E1:64_5_1_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E79">A79</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E12:64_5_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Q19</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1_1_2_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_5_1_1_2_2_1_2_3"/><span class="lab"><font color="Green" title="E79">A80</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font></span><br/>

<a NAME="E2:64_5_1_1_2_2_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">Q29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:64"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E30:64"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:64_5_1_1_2_2_1_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q29</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:64_5_1_1_2_2_1_2_3"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E77">A77</font></span></a>, <a class="txt" href="#E1:64_5_1_1_2_2_1_2_3"><span class="lab"><font color="Green" title="E79">A80</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><a NAME="E13:64_5_1_1_2_2_1_2"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E55:64"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="txt" href="#E56:64"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E57:64"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="txt" href="#E58:64"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E11:64_5_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E78">A78</font></span></a>, <a class="ref" href="jordan6.html#T59" target="_self" title="JORDAN6:th.59">Th59</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</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="E4:64_5_1_1_2"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:64_5"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</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>
<span class="kw">hence </span><a NAME="E60:64"/>
( ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> ) or ( <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P29</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P19</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>
