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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  holds <br/> <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b3">P</font>, <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <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">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  holds <br/> <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b1">P</font>, <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">fF</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">fB</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T11" title="TREAL_1:th.11">TREAL_1:11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  implies  <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font>, <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:110"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</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="c6">P</font>,<font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  or  <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font>, <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">F</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<a NAME="E4:110"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:110"/>
<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font>, <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f</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"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E7:110"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<font color="Maroon" title="c9">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E8:110"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">ff</font> = <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> 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"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">ff</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">ff</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive">b<sub>1</sub></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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )</span><br/>

<a NAME="E10:110"/>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> 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="borsuk_2.html#T12" title="BORSUK_2:th.12">BORSUK_2:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:110"/>
<font color="Maroon" title="c10">ff</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="#E7:110"><span class="lab"><font color="Green" title="E83">A3</font></span></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">b<sub>1</sub></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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E12:110"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E13:110"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:110_1"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">t9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t9</font> holds <br/><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">t9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t9</font> holds <br/><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">t9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">t9</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t9</font> implies <font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">t9</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:110_1_1"/><span class="lab"><font color="Green" title="E88">A8</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t9</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="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">t9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">ee</font> = <font color="Maroon" title="c11">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>;<br/>
<a NAME="E3:110_1_1"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D1" title="TREAL_1:def.1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2" title="TREAL_1:def.2">TREAL_1:def 2</a></span>;<br/>
<font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> = 
<font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ee</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">t9</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="txt" href="#E3:110_1_1"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="ref" href="treal_1.html#T10" title="TREAL_1:th.10">TREAL_1:10</a></span>
<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">t9</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E5:110_1_1"/>
<font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">t9</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">then </span><span class="lab"><font color="Green" title="E88">A10</font></span>: <font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>
<br/>.= 
1

;<br/>
<a NAME="E3:110_1"/>
1 is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E89">A11</font></span>: <font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1"><span class="lab"><font color="Green" title="E87">A7</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E5:110_1"/><span class="lab"><font color="Green" title="E90">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:110_1"/>
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">tt</font> = <font color="Maroon" title="c11">t</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></span>;<br/>
<a NAME="E8:110_1"/><span class="lab"><font color="Green" title="E91">A13</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:110_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E10:110_1"/><span class="lab"><font color="Green" title="E93">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>;<br/>
<a NAME="E11:110_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E9:110_1"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E12:110_1"/><span class="lab"><font color="Green" title="E95">A17</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E13:110_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E10:110_1"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c11">t</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E12:110_1"><span class="lab"><font color="Green" title="E95">A17</font></span></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c12">tt</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A19</font></span>: <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c11">t</font> = 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:110_1"><span class="lab"><font color="Green" title="E96">A18</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110"><span class="lab"><font color="Green" title="E84">A4</font></span></a></span>
;<br/>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E9:110_1"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c12">tt</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font> = 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c11">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110"><span class="lab"><font color="Green" title="E84">A4</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:110_1"/>
( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:110_1"><span class="lab"><font color="Green" title="E97">A19</font></span></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="E14:110"/><span class="lab"><font color="Green" title="E87">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:110"/>
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> )</span><br/>

<a NAME="E1:110_2"/><span class="lab"><font color="Green" title="E88">A21</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">t9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t9</font> holds <br/><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">t9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t9</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t9</font> holds <br/><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">t9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">t9</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t9</font> implies <font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t9</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:110_2_1"/><span class="lab"><font color="Green" title="E89">A22</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t9</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="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">ee</font> = <font color="Maroon" title="c12">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>;<br/>
<a NAME="E3:110_2_1"/><span class="lab"><font color="Green" title="E90">A23</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#D1" title="TREAL_1:def.1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2" title="TREAL_1:def.2">TREAL_1:def 2</a></span>;<br/>
<font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t</font> = 
<font color="Maroon" title="c5">fB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ee</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">t9</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_2_1"><span class="lab"><font color="Green" title="E89">A22</font></span></a>, <a class="txt" href="#E3:110_2_1"><span class="lab"><font color="Green" title="E90">A23</font></span></a>, <a class="ref" href="treal_1.html#T10" title="TREAL_1:th.10">TREAL_1:10</a></span>
<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">t9</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E5:110_2_1"/>
<font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t9</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>
<a NAME="E2:110_2"/>
<font color="Maroon" title="c11">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E3:110_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A24</font></span>: 
<font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_2"><span class="lab"><font color="Green" title="E88">A21</font></span></a></span>;<br/>
<a NAME="E4:110_2"/><span class="lab"><font color="Green" title="E90">A25</font></span>: 
1 is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E5:110_2"/><span class="lab"><font color="Green" title="E91">A26</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fB</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>;<br/>
<a NAME="E6:110_2"/><span class="lab"><font color="Green" title="E92">A27</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:110"><span class="lab"><font color="Green" title="E87">A20</font></span></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E7:110_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A28</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E5:110_2"><span class="lab"><font color="Green" title="E91">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E8:110_2"/><span class="lab"><font color="Green" title="E94">A29</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:110"><span class="lab"><font color="Green" title="E87">A20</font></span></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E9:110_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A30</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E5:110_2"><span class="lab"><font color="Green" title="E91">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:110_2"/><span class="lab"><font color="Green" title="E96">A31</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T4" title="JORDAN5B:th.4">JORDAN5B:4</a></span>;<br/>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,1 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E6:110_2"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_2"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="txt" href="#E4:110_2"><span class="lab"><font color="Green" title="E90">A25</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A32</font></span>: <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,1 = 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_2"><span class="lab"><font color="Green" title="E93">A28</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E10:110_2"><span class="lab"><font color="Green" title="E96">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D6" title="BORSUK_2:def.6">BORSUK_2:def 6</a></span>
;<br/>
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c5">fB</font>,<font color="Maroon" title="c4">fF</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fB</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E8:110_2"><span class="lab"><font color="Green" title="E94">A29</font></span></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E3:110_2"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110_2"><span class="lab"><font color="Green" title="E95">A30</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:110"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E10:110_2"><span class="lab"><font color="Green" title="E96">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D6" title="BORSUK_2:def.6">BORSUK_2:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:110_2"/>
( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110_2"><span class="lab"><font color="Green" title="E97">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:110"/>
for <font color="Olive">b<sub>1</sub></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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c10">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:110"><span class="lab"><font color="Green" title="E86">A6</font></span></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>
