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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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> <font color="Maroon" title="c1">n</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Olive" title="b1">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">p2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">p2</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>)</span><br/>







<div><span class="lab"><font color="Green" title="E27">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span> implies <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>) )</span><br/><span class="kw">assume </span><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<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>) &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">x1</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><span class="kw"> such that </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E30">A4</font></span>: <font color="Maroon" title="c5">y</font> = 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>
<br/>.= 
<font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D10" title="RLTOPSP1:def.10">RLTOPSP1:def 10</a></span>
;<br/><a NAME="E7:32_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E8:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y</font> as   <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> ;<br/><font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">p1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">p1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T41" title="EUCLID:th.41">EUCLID:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">p1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T26" title="EUCLID:th.26">EUCLID:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">p1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="euclid.html#T26" title="EUCLID:th.26">EUCLID:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">p1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T41" title="EUCLID:th.41">EUCLID:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T41" title="EUCLID:th.41">EUCLID:41</a></span>
<br/>.= 
<font color="Maroon" title="c8">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:32_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>) implies <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:32_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y1</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><span class="kw"> such that </span><br/><a NAME="E3:32_2"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/><span class="kw">set </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font>;<br/><a NAME="E4:32_2"/>
<font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</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> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E5:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:32_2"/>
<font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></span>;<br/><a NAME="E7:32_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D10" title="RLTOPSP1:def.10">RLTOPSP1:def 10</a></span>;<br/><a NAME="E8:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font></span>)</span>,<font color="Maroon" title="c6">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E9:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></span>;<br/><a NAME="E10:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:32_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></span>; <a class="txt" 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">hence </span><a NAME="E3:32"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> (<font color="Maroon" title="c2">p1</font>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">p2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mfold_2.html#K2" title="MFOLD_2:func.2">Plane</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
