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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c4">Y</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 ; <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"><font color="Maroon" title="c2">f</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><a NAME="E2:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T54" target="_self" title="PSCOMP_1:th.54">Th54</a></span>;<br/><a NAME="E3:70_1_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T139" title="FUNCT_1:th.139">FUNCT_1:139</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">Y</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</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="E2:70_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">X</font></span>)</span>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T54" target="_self" title="PSCOMP_1:th.54">Th54</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/>


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