<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>; <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">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">X</font></span><br/>



<a NAME="E1:69"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <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">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T35" target="_self" title="RLTOPSP1:th.35">Th35</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</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> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:69_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c4">u</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">a</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c3">x</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">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c4">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a></span>;<br/><a NAME="E5:69_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="rltopsp1.html#D10" target="_self" title="RLTOPSP1:def.10">Def10</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="rltopsp1.html#E54"><span class="lab"><font color="Green" title="E42">Lm9</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c5">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="rltopsp1.html#E54"><span class="lab"><font color="Green" title="E42">Lm9</font></span></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D10" target="_self" title="RLTOPSP1:def.10">Def10</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="E3:69"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="rltopsp1.html#K3" title="RLTOPSP1:func.3">transl</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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>
