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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">r</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<a NAME="E1:96"/><i><font color="Green" title="E59">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</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>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T48" target="_self" title="RLTOPSP1:th.48">Th48</a></i>;<br/>
<a NAME="E2:96"/><i><font color="Green" title="E60">A2</font></i>: 
 <a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E75"><i><font color="Green" title="E57">Lm14</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>consider </b><font color="Maroon" title="c4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:96_1"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 <b>and </b><br/><a NAME="E3:96_1"/><i><font color="Green" title="E62">A4</font></i>: 
<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#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</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>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="rltopsp1.html#E2:96_1"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/><a NAME="E5:96_1"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:96_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="rltopsp1.html#D13" target="_self" title="RLTOPSP1:def.13">Def13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</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#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</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>
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E2:96"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c5">u</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:96"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E3:96_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:96_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </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>
<b>by </b><i><a class="ref" href="rltopsp1.html#D13" target="_self" title="RLTOPSP1:def.13">Def13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:96"/>
<span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> <font color="Maroon" title="c2">r</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#K4" title="RLTOPSP1:func.4">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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