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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:102_1_1_1"/>
(  not <font color="Maroon" title="c1">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or <font color="Maroon" title="c1">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:102_1_1_1_1"/>
not <font color="Maroon" title="c1">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c1">n</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">n</font> = <font color="Maroon" title="c1">n</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c2">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">J</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> = 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="c2">n</font></span>)</span>;<br/><a NAME="E3:102_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#T28" title="EUCLID_9:th.28">EUCLID_9:28</a></span>;<br/><a NAME="E4:102_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</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="c2">n</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> =  <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c2">n</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#T28" title="EUCLID_9:th.28">EUCLID_9:28</a></span>;<br/><a NAME="E6:102_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
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="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c9">P</font> =  <a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font>;<br/><a NAME="E1:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E3:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/><a NAME="E4:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
for <font color="Olive" title="b1">a</font>, <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="c2">n</font></span>)</span> holds  <font color="Maroon" title="c10">f</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</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="c2">n</font></span>)</span>; <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="c10">f</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c10">f</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font> = 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="toprealc.html#K11" title="TOPREALC:func.11">(#)</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealc.html#D5" target="_self" title="TOPREALC:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="toprealc.html#K11" title="TOPREALC:func.11">(#)</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E4:102_1_1_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="yellow17.html#T8" title="YELLOW17:th.8">YELLOW17:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</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">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c1">n</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span> holds  <font color="Maroon" title="c11">f1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c12">f1</font> = <font color="Maroon" title="c11">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">f2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E9:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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="c2">n</font></span>)</span> holds  <font color="Maroon" title="c13">f2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c12">f1</font>, <font color="Maroon" title="c15">f2</font> = <font color="Maroon" title="c13">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a></span>;<br/><a NAME="E11:102_1_1_1_1_1"/>
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"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c18">p1</font>, <font color="Maroon" title="c19">p2</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="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:102_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">B1</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c17">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">B2</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">W</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c22">W</font>
; <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="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<a NAME="E4:102_1_1_1_1_1_2"/>
( <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c17">r</font> &amp; <font color="Maroon" title="c19">p2</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T17" title="TOPGEN_5:th.17">TOPGEN_5:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:102_1_1_1_1_1_2"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <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="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> or <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E6:102_1_1_1_1_1_2"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font>
 ; <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="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E8:102_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">and </span><br/><a NAME="E9:102_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">x1</font>, <font color="Maroon" title="c26">x2</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="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:102_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<a NAME="E12:102_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
( <font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a></span>;<br/>
<a NAME="E13:102_1_1_1_1_1_2"/>
<font color="Maroon" title="c25">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E11:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E14:102_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<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="c25">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c18">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/>
<a NAME="E15:102_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c18">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E16:102_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c18">p1</font></span>)</span> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E17:102_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">p1</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c25">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c18">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:102_1_1_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="real_ns1.html#T8" title="REAL_NS1:th.8">REAL_NS1:8</a></span>;<br/>
<a NAME="E18:102_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E11:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E12:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E14:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:102_1_1_1_1_1_2"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:102_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</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><a NAME="E12:102_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c14">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="tops_4.html#T21" title="TOPS_4:th.21">TOPS_4:21</a></span>;<br/><a NAME="E13:102_1_1_1_1_1"/>
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"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c18">p1</font>, <font color="Maroon" title="c19">p2</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="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:102_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">B1</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">B2</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p2</font>,<font color="Maroon" title="c17">r</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">W</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p2</font>,<font color="Maroon" title="c17">r</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c22">W</font>
; <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="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<a NAME="E4:102_1_1_1_1_1_3"/>
( <font color="Maroon" title="c18">p1</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c19">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p2</font>,<font color="Maroon" title="c17">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T17" title="TOPGEN_5:th.17">TOPGEN_5:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:102_1_1_1_1_1_3"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <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="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font> or <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E6:102_1_1_1_1_1_3"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c22">W</font>
 ; <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="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E8:102_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">and </span><br/><a NAME="E9:102_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">x1</font>, <font color="Maroon" title="c26">x2</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="c2">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:102_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/>
<a NAME="E12:102_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
( <font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a></span>;<br/>
<a NAME="E13:102_1_1_1_1_1_3"/>
<font color="Maroon" title="c26">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p2</font>,<font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E11:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E14:102_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<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="c26">x2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></span>;<br/>
<a NAME="E15:102_1_1_1_1_1_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">x2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T3" title="EUCLID:th.3">EUCLID:3</a></span>;<br/>
<a NAME="E16:102_1_1_1_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">x2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">x2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">p2</font></span>)</span> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>;<br/>
<a NAME="E17:102_1_1_1_1_1_3"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">x2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">p2</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c26">x2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:102_1_1_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="real_ns1.html#T8" title="REAL_NS1:th.8">REAL_NS1:8</a></span>;<br/>
<a NAME="E18:102_1_1_1_1_1_3"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="#E11:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="#E12:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E14:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:102_1_1_1_1_1_3"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:102_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="ref" href="rcomp_1.html#T8" title="RCOMP_1:th.8">RCOMP_1:8</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><a NAME="E14:102_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">f2</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="tops_4.html#T21" title="TOPS_4:th.21">TOPS_4:21</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw"> such that </span><br/><a NAME="E16:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A24</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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><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="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</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="c14">f1</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</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> &amp; <font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c16">g</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">r2</font>
 <span class="kw">and </span><br/><a NAME="E17:102_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c16">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="txt" href="#E12:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="ref" href="jgraph_2.html#T35" title="JGRAPH_2:th.35">JGRAPH_2:35</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">a</font>, <font color="Maroon" title="c18">b</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="c2">n</font></span>)</span>; <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="c10">f</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font></span><br/><a NAME="E1:102_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
( <font color="Maroon" title="c14">f1</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c15">f2</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E9:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c10">f</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">a</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="toprealc.html#K4" title="TOPREALC:func.4">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">g</font> <a href="toprealc.html#K5" title="TOPREALC:func.5">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E1:102_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E53">A26</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">g</font> <a href="toprealc.html#K6" title="TOPREALC:func.6">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font>

; <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="E19:102_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> </span>)</span>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">F</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="#E17:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E1:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E3:102_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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><a NAME="E8:102_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</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="waybel18.html#T7" title="WAYBEL18:th.7">WAYBEL18:7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:102_1_1_1_1"/>
 <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c1">n</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="#E3:102_1_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E4:102_1_1_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:102_1_1_1_2"/>
<font color="Maroon" title="c1">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c1">n</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:102_1_1_1_2"/>
 <a href="toprealc.html#K16" title="TOPREALC:func.16">TIMES</a> <font color="Maroon" title="c1">n</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="toprealc.html#T53" target="_self" title="TOPREALC:th.53">Th53</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></div><span class="kw">end;</span></div>

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