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<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> is <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a> implies <font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )</span><br/>

<b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon" title="c6">f</font> is <a href="jordan24.html#V1" title="JORDAN24:attr.1">isometric</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/>

<b>then consider </b><font color="Maroon" title="c7">g</font> being  <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c5">M</font><b> such that </b><br/><a NAME="E3:5_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="jordan24.html#D2" target="_self" title="JORDAN24:def.2">Def2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font>  ex <font color="Olive">b<sub>2</sub></font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">W</font> st <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>let </b><font color="Maroon" title="c9">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">W</font> st <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font></span><br/>



<b>reconsider </b><font color="Maroon" title="c10">fw</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font>, <font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c8">W</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">M</font> ;<br/>
<a NAME="E5:5_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">r</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> <b> such that </b><br/><a NAME="E7:5_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">fw</font>,<font color="Maroon" title="c12">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font> )
 <b>by </b><i><a class="ref" href="goboard6.html#T7" title="GOBOARD6:th.7">GOBOARD6:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">H</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">r</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">W</font> <b>by </b><i><a class="txt" href="#E7:5_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="goboard6.html#T94" title="GOBOARD6:th.94">GOBOARD6:94</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font></span><br/>

<b>thus </b><a NAME="E9:5_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">H</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font> )</span><br/>

<b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:5_1_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c15">b</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> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">H</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c15">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c5">M</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:5_1_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">b'</font> = <font color="Maroon" title="c16">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">M</font> ;<br/>
<a NAME="E6:5_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c17">b'</font>,<font color="Maroon" title="c11">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="#E3:5_1_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E7:5_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b'</font></span>)</span>,<font color="Maroon" title="c10">fw</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="#E3:5_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="vectmetr.html#D3" title="VECTMETR:def.3">VECTMETR:def 3</a></i>;<br/>
<a NAME="E8:5_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c10">fw</font>,<font color="Maroon" title="c12">r</font>
 
<b>by </b><i><a class="txt" href="#E3:5_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E3:5_1_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:5_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="#E7:5_1_1"><i><font color="Green" title="E5">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
