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<div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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="c13">r</font> be  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c14">x</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="c12">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">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="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font></span>)</span><br/> for <font color="Olive" title="b2">b</font> being  <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="c12">n</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c15">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c16">U</font> =  <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font>;<br/>
<b>set </b><font color="Maroon" title="c17">C</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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="c12">n</font></span>)</span>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> $1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>;<br/>
<a NAME="E2:51"/><i><font color="Green" title="E30">A2</font></i>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green" title="E31">A3</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green" title="E32">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1
 
<b>by </b><i><a class="ref" href="toprealb.html#T9" target="_self" title="TOPREALB:th.9">Th9</a></i>;<br/>
<a NAME="E5:51"/><i><font color="Green" title="E33">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T9" target="_self" title="TOPREALB:th.9">Th9</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green" title="E34">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:51"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E8:51"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c18">b</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><br/>

<b>assume </b><a NAME="E2:51_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c18">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">a</font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>;<br/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T13" title="JORDAN2C:th.13">JORDAN2C:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E2:51"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="toprns_1.html#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">r</font>
<b>by </b><i><a class="txt" href="toprealb.html#E5:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="toprealb.html#E2:51_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>
;<br/>
<a NAME="E5:51_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<b>then </b><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">c</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
;<br/>
<b>hence </b><a NAME="E7:51_1"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="toprealb.html#E6:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="toprealb.html#E5:51_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E9:51"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c15">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_2"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c18">a</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="c15">f</font>
 <b>and </b><br/><a NAME="E2:51_2"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c19">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="c15">f</font>
 <b>and </b><br/><a NAME="E3:51_2"/><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">a1</font> = <font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c21">b1</font> = <font color="Maroon" title="c19">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="toprealb.html#E6:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="toprealb.html#E1:51_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="toprealb.html#E2:51_2"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/>
<a NAME="E5:51_2"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c20">a1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="toprealb.html#E1:51_2"><i><font color="Green" title="E37">A11</font></i></a></i>;<br/>
<a NAME="E6:51_2"/>
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c21">b1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="toprealb.html#E2:51_2"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/>
<a NAME="E7:51_2"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c20">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c21">b1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:51_2"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="toprealb.html#E5:51_2"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>;<br/>
<a NAME="E8:51_2"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c20">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c21">b1</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>;<br/>
<b>hence </b><a NAME="E9:51_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font>
 <b>by </b><i><a class="ref" href="euclid.html#T38" title="EUCLID:th.38">EUCLID:38</a></i>;<br/>


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

<a NAME="E10:51"/><i><font color="Green" title="E37">A15</font></i>: 
for <font color="Olive" title="b1">u</font> being  <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="c12">n</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being  <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="c12">n</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c18">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span><b> such that </b><br/><a NAME="E12:51"/><i><font color="Green" title="E38">A16</font></i>: 
for <font color="Olive" title="b1">x</font> being  <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="c12">n</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c18">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="toprealb.html#E10:51"><i><font color="Green" title="E37">A15</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c19">f1</font> =  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c20">f2</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font>;<br/>
<a NAME="E13:51"/>
for <font color="Olive" title="b1">p</font> being  <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="c12">n</font></span>)</span> holds  <font color="Maroon" title="c18">F</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">p</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="c12">n</font></span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c21">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c21">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c18">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font>
<b>by </b><i><a class="txt" href="toprealb.html#E12:51"><i><font color="Green" title="E38">A16</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E14:51"/><b>then </b><i><font color="Green" title="E39">A17</font></i>: 
<font color="Maroon" title="c18">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T17" title="TOPALG_1:th.17">TOPALG_1:17</a></i>;<br/>
<a NAME="E15:51"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:51"/><b>then </b><i><font color="Green" title="E40">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E17:51"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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="c15">f</font> holds <br/><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_4"/><i><font color="Green" title="E41">A19</font></i>: 
<font color="Maroon" title="c21">a</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="c15">f</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">a</font> as   <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="c12">n</font></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E1:51_4"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c22">y</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="toprealb.html#E1:51_4"><i><font color="Green" title="E41">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">y</font>
<b>by </b><i><a class="txt" href="toprealb.html#E12:51"><i><font color="Green" title="E38">A16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font>
<b>by </b><i><a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="toprealb.html#E1:51_4"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E18:51"/><b>then </b>
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="toprealb.html#E6:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="toprealb.html#E16:51"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E19:51"/>
<font color="Maroon" title="c15">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:51"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="borsuk_4.html#T69" title="BORSUK_4:th.69">BORSUK_4:69</a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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="c12">n</font></span>)</span>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default">$1 <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>;<br/>
<a NAME="E20:51"/><i><font color="Green" title="E41">A20</font></i>: 
for <font color="Olive" title="b1">u</font> being  <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="c12">n</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being  <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="c12">n</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c21">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span><b> such that </b><br/><a NAME="E22:51"/><i><font color="Green" title="E42">A21</font></i>: 
for <font color="Olive" title="b1">a</font> being  <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="c12">n</font></span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c21">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="toprealb.html#E20:51"><i><font color="Green" title="E41">A20</font></i></a>);<br/></i>
<a NAME="E23:51"/><i><font color="Green" title="E43">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:51"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:51"/><b>then </b><i><font color="Green" title="E44">A23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E26:51"/>
for <font color="Olive" title="b1">p</font> being  <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="c12">n</font></span>)</span> holds  <font color="Maroon" title="c21">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">p</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="c12">n</font></span>)</span>;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44" title="EUCLID:th.44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c22">p</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></i>
<br/>.= 
<font color="Maroon" title="c21">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font>
<b>by </b><i><a class="txt" href="toprealb.html#E22:51"><i><font color="Green" title="E42">A21</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:51"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
<font color="Maroon" title="c21">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topalg_1.html#T17" title="TOPALG_1:th.17">TOPALG_1:17</a></i>;<br/>
<a NAME="E28:51"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:51_6"/><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c22">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c22">a</font> as   <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="c12">n</font></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E1:51_6"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">ff</font> = <font color="Maroon" title="c15">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>set </b><font color="Maroon" title="c25">e</font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>;<br/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T13" title="JORDAN2C:th.13">JORDAN2C:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E2:51"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="toprns_1.html#T8" title="TOPRNS_1:th.8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">r</font>
<b>by </b><i><a class="txt" href="toprealb.html#E5:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="toprealb.html#E1:51_6"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>
;<br/>
<a NAME="E5:51_6"/><b>then </b><i><font color="Green" title="E47">A26</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c12">n</font></span>)</span>,1
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<b>then </b><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
;<br/>
<a NAME="E7:51_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="toprealb.html#E6:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="toprealb.html#E9:51"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="toprealb.html#E5:51_6"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:51"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="toprealb.html#E9:51"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c21">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="toprealb.html#E22:51"><i><font color="Green" title="E42">A21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<b>by </b><i><a class="txt" href="toprealb.html#E5:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="toprealb.html#E1:51_6"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E29:51"/><b>then </b>
<font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E5:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="toprealb.html#E23:51"><i><font color="Green" title="E43">A22</font></i></a>, <a class="txt" href="toprealb.html#E25:51"><i><font color="Green" title="E44">A23</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E30:51"/>
<font color="Maroon" title="c15">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E27:51"><i><font color="Green" title="E45">A24</font></i></a>, <a class="ref" href="borsuk_4.html#T69" title="BORSUK_4:th.69">BORSUK_4:69</a></i>;<br/>


</div>
