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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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> <font color="Maroon" title="c1">n</font></span>)</span> holds  <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b4">q2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span><br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">q2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span><br/>









<a NAME="E1:43"/><i><font color="Green" title="E32">A1</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T40" target="_self" title="EUCLID_2:th.40">Th40</a></i>;<br/>
<a NAME="E2:43"/><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T40" target="_self" title="EUCLID_2:th.40">Th40</a></i>;<br/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">q2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T40" target="_self" title="EUCLID_2:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="#E2:43"><i><font color="Green" title="E33">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:43"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><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">+</a> <font color="Maroon" title="c3">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">q2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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