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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>let </b><font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c4">w2</font>, <font color="Maroon" title="c5">w3</font>, <font color="Maroon" title="c6">w4</font>, <font color="Maroon" title="c7">w5</font>, <font color="Maroon" title="c8">w6</font>, <font color="Maroon" title="c9">w7</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>;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c6">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c7">w5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">w6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">w7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">w3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">w3</font>,<font color="Maroon" title="c6">w4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c6">w4</font>,<font color="Maroon" title="c7">w5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c7">w5</font>,<font color="Maroon" title="c8">w6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c8">w6</font>,<font color="Maroon" title="c9">w7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
 ex <font color="Olive" title="b1">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c6">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th45</a></i>;<br/>
<a NAME="E3:50"/>
 ex <font color="Olive" title="b1">h4</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h4</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c6">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c9">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:50"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E4:50"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><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> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp; <font color="Maroon" title="c9">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:50"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="jordan2c.html#E2:50"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="jordan2c.html#T43" target="_self" title="JORDAN2C:th.43">Th43</a></i>;<br/>


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