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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">q3</font>, <font color="Maroon" title="c5">q4</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q</font> where <font color="Olive" title="b2">q</font> is   <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> : <font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  holds <br/><font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">Q</font>) ) &amp; (  for <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q</font> where <font color="Olive" title="b2">q</font> is   <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> : <font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  holds <br/><font color="Maroon" title="c5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">Q</font>) ) implies <font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q4</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1_2"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( (  for <font color="Olive" title="b1">Q1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Olive" title="b1">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q1</font> where <font color="Olive" title="b2">q1</font> is   <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> : <font color="Olive" title="b2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  holds <br/><font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">Q1</font>) ) &amp; (  for <font color="Olive" title="b1">Q2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Olive" title="b1">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q2</font> where <font color="Olive" title="b2">q2</font> is   <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> : <font color="Olive" title="b2">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  holds <br/><font color="Maroon" title="c5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">Q2</font>) ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q4</font></span><br/>

<a NAME="E2:14_1_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">q1</font> where <font color="Olive" title="b1">q1</font> is   <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> : <font color="Olive" title="b1">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q1</font> where <font color="Olive" title="b1">q1</font> is   <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> : <font color="Olive" title="b1">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q1</font> where <font color="Olive" title="b1">q1</font> is   <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> : <font color="Olive" title="b1">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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></span><br/>

<a NAME="E2:14_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</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 <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:14_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Q3</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <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> : <font color="Olive" title="b1">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E4:14_1_2"/>
<font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<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="c6">Q3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1_2"/>
<font color="Maroon" title="c4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
