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<span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as   <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> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E2:24_1_1"/>
<font color="Maroon" title="c7">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g</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> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw"> such that </span><br/><a NAME="E4:24_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">r1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c7">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">r1</font>
 <span class="kw">and </span><br/><a NAME="E5:24_1_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c8">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T33" title="JGRAPH_2:th.33">JGRAPH_2:33</a></span>;<br/>
<a NAME="E6:24_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K1" title="TOPALG_5:func.1">ExtendInt</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K1" title="TOPALG_5:func.1">ExtendInt</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> = 
<font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K1" title="TOPALG_5:func.1">ExtendInt</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</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="E7:24_1_1"/>
 <a href="topalg_5.html#K1" title="TOPALG_5:func.1">ExtendInt</a> <font color="Maroon" title="c6">r</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <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/>


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