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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  holds <br/><font color="Olive" title="b1">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> </span><br/><b>let </b><font color="Maroon" title="c2">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">p</font>, <font color="Olive" title="b2">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> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  holds <br/><font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> </span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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"> ( <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  implies <font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/>







<b>assume </b><a NAME="E1:68"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> </span><br/>

<a NAME="E2:68"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_2.html#D3" target="_self" title="EUCLID_2:def.3">Def3</a></i>;<br/>
<a NAME="E3:68"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E4:68"/><b>then </b>
<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">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c4">q</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_2.html#T41" target="_self" title="EUCLID_2:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E5:68"/>
<font color="Maroon" title="c2">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 <b>by </b><i><a class="ref" href="euclid_2.html#D3" target="_self" title="EUCLID_2:def.3">Def3</a></i>; <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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