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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds   <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Olive" title="b1">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></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>



<span class="kw">thus </span><a NAME="E1:93"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">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#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:93"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 ; <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="c3">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#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><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> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><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> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D13" target="_self" title="RLTOPSP1:def.13">Def13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:93"/>
<font color="Maroon" title="c3">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#K4" title="RLTOPSP1:func.4">mlt</a> (<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</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/>


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