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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">c1</font> holds <br/>( <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  iff  ex <font color="Olive" title="b4">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b4">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b7">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b7">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">t</font> &amp; <font color="Olive" title="b4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b7">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">t</font> ) ) &amp; (  for <font color="Olive" title="b7">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="Olive" title="b4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Olive" title="b3">S</font></span>)</span>,<font color="Olive" title="b7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a</font> &amp; <font color="Olive" title="b4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Olive" title="b3">S</font></span>)</span>,<font color="Olive" title="b7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">b</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</font> be   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font> &amp; <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  iff  ex <font color="Olive" title="b2">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b5">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t</font> &amp; <font color="Olive" title="b2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t</font> ) ) &amp; (  for <font color="Olive" title="b5">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="Olive" title="b2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Olive" title="b1">S</font></span>)</span>,<font color="Olive" title="b5">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Olive" title="b1">S</font></span>)</span>,<font color="Olive" title="b5">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> implies ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  iff  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:67"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  iff  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) ) )</span><br/>

<a NAME="E2:67"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" onmouseover="rs('xxreal_2/T40')" onmouseout="rh()">XXREAL_2:40</a></span>;<br/>
<a NAME="E3:67"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="topalg_6.html#T27" target="_self" onmouseover="rs('topalg_6/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E4:67"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E5:67"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E6:67"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E7:67"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:67"/>
( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  implies  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) ) implies <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1"/>
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> or  not  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font><span class="kw"> such that </span><br/><a NAME="E3:67_1_1_1"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c8">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a></span>;<br/><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#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E5:67_1_1_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
( <font color="Maroon" title="c9">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  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="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</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">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">f1</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1);<br/><span class="kw">set </span><font color="Maroon" title="c11">f2</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1);<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">S1</font> = <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font> as   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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/><a NAME="E7:67_1_1_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="topmetr.html#T19" onmouseover="rs('topmetr/T19')" onmouseout="rh()">TOPMETR:19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">f1</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) 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 <font color="Maroon" title="c12">S1</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="txt" href="#E7:67_1_1_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="borsuk_6.html#T34" onmouseover="rs('borsuk_6/T34')" onmouseout="rh()">BORSUK_6:34</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">f2</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) 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="borsuk_6.html#T34" onmouseover="rs('borsuk_6/T34')" onmouseout="rh()">BORSUK_6:34</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">h</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="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">h</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="c13">f1</font>,<font color="Maroon" title="c14">f2</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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c16">h</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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> st <br/>( <font color="Maroon" title="c16">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b3">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> ) ) &amp; (  for <font color="Olive" title="b3">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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c5">a</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c16">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">b</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> ) ) &amp; (  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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E11:67_1_1_1"/>
<font color="Maroon" title="c16">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> ) ) &amp; (  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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> ) ) )</span><br/><a NAME="E12:67_1_1_1"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E13:67_1_1_1"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E53">A15</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/><span class="lab"><font color="Green" title="E54">A16</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
1

;<br/><a NAME="E16:67_1_1_1"/><span class="lab"><font color="Green" title="E55">A17</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/><a NAME="E17:67_1_1_1"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67_1_1_1"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E18:67_1_1_1"/><span class="lab"><font color="Green" title="E57">A19</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67_1_1_1"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E19:67_1_1_1"/><span class="lab"><font color="Green" title="E58">A20</font></span>: 
<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E20:67_1_1_1"/>
 for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> )</span><br/>

<a NAME="E1:67_1_1_1_3"/><span class="lab"><font color="Green" title="E59">A21</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>;<br/>
<a NAME="E2:67_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E17:67_1_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E3:67_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E4:67_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E18:67_1_1_1"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E5:67_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E6:67_1_1_1_3"/><span class="lab"><font color="Green" title="E62">A24</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E7:67_1_1_1_3"/><span class="lab"><font color="Green" title="E63">A25</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_1_3"><span class="lab"><font color="Green" title="E59">A21</font></span></a>, <a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E8:67_1_1_1_3"/>
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_1_1_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/>
<a NAME="E9:67_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A26</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:67_1_1_1_3"/><span class="lab"><font color="Green" title="E65">A27</font></span>: 
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c17">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_1_1_1_3"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A28</font></span>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1_1_3"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:67_1_1_1"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font>

;<br/>
<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_1_3"><span class="lab"><font color="Green" title="E60">A22</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E17:67_1_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<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"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_1_1_1"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_1_3"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E16:67_1_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T2" target="_self" onmouseover="rs('topalg_6/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67_1_1_1_3"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="txt" href="#E9:67_1_1_1_3"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1_3"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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="c17">t</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</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="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E18:67_1_1_1"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<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"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:67_1_1_1"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_1_3"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E16:67_1_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T2" target="_self" onmouseover="rs('topalg_6/T2')" onmouseout="rh()">Th2</a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67_1_1_1_3"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="txt" href="#E9:67_1_1_1_3"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E21:67_1_1_1"/>
 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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<a NAME="E1:67_1_1_1_4"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font> <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="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E2:67_1_1_1_4"/><span class="lab"><font color="Green" title="E60">A30</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font> <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="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E12:67_1_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E3:67_1_1_1_4"/>
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
;<br/>
<a NAME="E4:67_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A31</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:67_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E4:67_1_1_1_4"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E1:67_1_1_1_4"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:67_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E7:67_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E4:67_1_1_1_4"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E2:67_1_1_1_4"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E8:67_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A33</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E9:67_1_1_1_4"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c17">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E64">A34</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font>

;<br/>
<span class="lab"><font color="Green" title="E65">A35</font></span>: <font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/>
<span class="lab"><font color="Green" title="E66">A36</font></span>: <font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:67_1_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" onmouseover="rs('xcmplx_1/T75')" onmouseout="rh()">XCMPLX_1:75</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:67_1_1_1"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_1_4"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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> (<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">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="#E4:67_1_1_1_4"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E1:67_1_1_1_4"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<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"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1_1_4"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E11:67_1_1_1_4"><span class="lab"><font color="Green" title="E65">A35</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_1_1_1_4"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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> (<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">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="#E4:67_1_1_1_4"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E2:67_1_1_1_4"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<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"><font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<font color="Maroon" title="c17">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1_1_4"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E12:67_1_1_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_2"/>
 not  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/><div class="add"><a NAME="E2:67_1_1_2"/><span class="kw">then </span>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:67_1_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T27" target="_self" onmouseover="rs('topalg_6/T27')" onmouseout="rh()">Th27</a></span>;<br/><a NAME="E4:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A37</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T17" onmouseover="rs('xxreal_1/T17')" onmouseout="rh()">XXREAL_1:17</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">a</font> =  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">f</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p1">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span>;<br/><a NAME="E5:67_1_1_2"/><span class="lab"><font color="Green" title="E48">A38</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 <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> )</span><br/>

<a NAME="E1:67_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A39</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/>
<a NAME="E2:67_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A40</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">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> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></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="txt" href="#E4:67"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E3:67_1_1_2_1"/><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">t</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> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></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="txt" href="#E5:67"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E4:67_1_1_2_1"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1_2_1"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E5:67_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E4:67_1_1_2"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) = 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E2:67_1_1_2_1"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">t</font>,1) = 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">t</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_2_1"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c3">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="topalg_6.html#T35" target="_self" onmouseover="rs('topalg_6/T35')" onmouseout="rh()">Th35</a></span>
<br/>.= 
<font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E4:67_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E4:67_1_1_2"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:67_1_1_2"/>
 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/> ( <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font> )</span><br/>

<a NAME="E1:67_1_1_2_2"/><span class="lab"><font color="Green" title="E49">A44</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</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="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E2:67_1_1_2_2"/><span class="lab"><font color="Green" title="E50">A45</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</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="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E3:67_1_1_2_2"/><span class="lab"><font color="Green" title="E51">A46</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">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> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></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="txt" href="#E1:67_1_1_2_2"><span class="lab"><font color="Green" title="E49">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E4:67_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A47</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">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> <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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></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="txt" href="#E2:67_1_1_2_2"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font>) = 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_1_2_2"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font>) = 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <span class="p2">(<span class="default"><a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c7">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_2_2"><span class="lab"><font color="Green" title="E52">A47</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:67_1_1_2"/>
 ex <font color="Olive" title="b1">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> ) ) &amp; (  for <font color="Olive" title="b4">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="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1_1_2"><span class="lab"><font color="Green" title="E48">A38</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c5">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><a NAME="E10:67"/><span class="lab"><font color="Green" title="E46">A48</font></span>: 
( <font color="Maroon" title="c5">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span> holds <br/> ( <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> ) ) &amp; (  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="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> ) ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a> </span><br/>

<a NAME="E11:67"/><span class="lab"><font color="Green" title="E47">A49</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</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="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E12:67"/><span class="lab"><font color="Green" title="E48">A50</font></span>: 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</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="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E49">A51</font></span>: <font color="Maroon" title="c6">a</font> = 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
;<br/>
<font color="Maroon" title="c7">b</font> = 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
 <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:67"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="topalg_6.html#T29" target="_self" onmouseover="rs('topalg_6/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A52</font></span>: <font color="Maroon" title="c6">a</font> = 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
 <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c3">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
;<br/>
<font color="Maroon" title="c7">b</font> = 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
 <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c3">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E12:67"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p2</font> = <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:67"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="ref" href="topalg_6.html#T29" target="_self" onmouseover="rs('topalg_6/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f1</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">f2</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">S1</font> = <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font> as   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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<a NAME="E20:67"/><span class="lab"><font color="Green" title="E51">A53</font></span>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="topmetr.html#T19" onmouseover="rs('topmetr/T19')" onmouseout="rh()">TOPMETR:19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f1</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></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>,<font color="Maroon" title="c12">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E20:67"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="ref" href="borsuk_6.html#T34" onmouseover="rs('borsuk_6/T34')" onmouseout="rh()">BORSUK_6:34</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f2</font> =  <a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) 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="borsuk_6.html#T34" onmouseover="rs('borsuk_6/T34')" onmouseout="rh()">BORSUK_6:34</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c5">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="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c5">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E24:67"/><span class="lab"><font color="Green" title="E52">A54</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E25:67"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A56</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/>
<span class="lab"><font color="Green" title="E55">A57</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
1

;<br/>
<a NAME="E28:67"/><span class="lab"><font color="Green" title="E56">A58</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E29:67"/><span class="lab"><font color="Green" title="E57">A59</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E30:67"/>
 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="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c16">h</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="c7">b</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<a NAME="E1:67_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E28:67"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E2:67_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A60</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E3:67_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E29:67"><span class="lab"><font color="Green" title="E57">A59</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E4:67_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A61</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E5:67_8"/><span class="lab"><font color="Green" title="E60">A62</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E6:67_8"/><span class="lab"><font color="Green" title="E61">A63</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E7:67_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E5:67_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E8:67_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A64</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E9:67_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c17">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> <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">f1</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="c14">f2</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="txt" href="#E6:67_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E10:67_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A65</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c17">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="c13">f1</font>,<font color="Maroon" title="c14">f2</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="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A66</font></span>: <font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E65">A67</font></span>: <font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p5">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>
;<br/>
<a NAME="E13:67_8"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c17">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E66">A68</font></span>: <font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T35" onmouseover="rs('borsuk_6/T35')" onmouseout="rh()">BORSUK_6:35</a></span>
<br/>.= 
<font color="Maroon" title="c17">t</font>

;<br/>
<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_8"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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="c17">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E28:67"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t</font>,1) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">t</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_8"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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="c17">t</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</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="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E29:67"><span class="lab"><font color="Green" title="E57">A59</font></span></a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:67"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c3">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">p2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E24:67"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c17">t</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">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="#E5:67_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67_8"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E14:67_8"><span class="lab"><font color="Green" title="E66">A68</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c17">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c17">t</font>) = 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c17">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_8"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</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="c17">t</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c13">f1</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="c14">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">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="#E6:67_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E25:67"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="funct_3.html#D8" onmouseover="rs('funct_3/D8')" onmouseout="rh()">FUNCT_3:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <font color="Maroon" title="c4">S</font></span>)</span>,<font color="Maroon" title="c17">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_8"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E14:67_8"><span class="lab"><font color="Green" title="E66">A68</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:67"/><span class="kw">then </span>
<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:67"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R2" title="TOPALG_6:pred.2">are_homotopic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
