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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b5">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font><br/>  for <font color="Olive" title="b7">R</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b6">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b7">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b7">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" 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">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b2">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font><br/>  for <font color="Olive" title="b3">R</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" 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">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font><br/>  for <font color="Olive" title="b2">R</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b2">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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">R</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/> <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">R</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">R</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</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">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</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="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">F</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f</font> =  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>;<br/>
<a NAME="E4:67"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="borsuk_6.html#T42" target="_self" onmouseover="rs('borsuk_6/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E5:67"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="borsuk_6.html#T42" target="_self" onmouseover="rs('borsuk_6/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E6:67"/>
 for <font color="Olive" title="b1">x</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  <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<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">let </span><font color="Maroon" title="c12">x</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"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<a NAME="E1:67_1"/>
<font color="Maroon" title="c12">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
;<br/>
<a NAME="E2:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A6</font></span>: 
<font color="Maroon" title="c12">x</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> <a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</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/>
<span class="lab"><font color="Green" title="E59">A7</font></span>: <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="borsuk_6.html#D4" target="_self" onmouseover="rs('borsuk_6/D4')" onmouseout="rh()">Def4</a>, <a class="ref" href="borsuk_6.html#T42" target="_self" onmouseover="rs('borsuk_6/T42')" onmouseout="rh()">Th42</a>, <a class="ref" href="borsuk_6.html#T49" target="_self" onmouseover="rs('borsuk_6/T49')" onmouseout="rh()">Th49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><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_2"/>
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 or ( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 &amp; <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4 ) or <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_2_1"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ; <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"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T6" target="_self" onmouseover="rs('borsuk_6/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E3:67_1_2_1"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E4:67_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A9</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">z</font> = 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E1:67_1_2_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="borsuk_6.html#D7" target="_self" onmouseover="rs('borsuk_6/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E4:67_1_2_1"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E1:67_1_2_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E1:67_1_2_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:67_1_2_1"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</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><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_2_2"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 &amp; <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4 )
 ; <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"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><a NAME="E2:67_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E3:67_1_2_2"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E4:67_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <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="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_2_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">z</font> = 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1_2_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="borsuk_6.html#T3" target="_self" onmouseover="rs('borsuk_6/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E7:67_1_2_2"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_2_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="borsuk_6.html#T4" target="_self" onmouseover="rs('borsuk_6/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E9:67_1_2_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:67_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E1:67_1_2_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="borsuk_6.html#D7" target="_self" onmouseover="rs('borsuk_6/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E3:67_1_2_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">z</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E7:67_1_2_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">w</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E10:67_1_2_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E1:67_1_2_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:67_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</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><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:67_1_2_3"/><span class="lab"><font color="Green" title="E60">A15</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
 ; <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"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">w</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T4" target="_self" onmouseover="rs('borsuk_6/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:67_1_2_3"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_3"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><a NAME="E4:67_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A16</font></span>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_3"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="borsuk_6.html#T4" target="_self" onmouseover="rs('borsuk_6/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:67_1_2_3"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_2_3"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="txt" href="#E1:67_1_2_3"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="borsuk_6.html#D7" target="_self" onmouseover="rs('borsuk_6/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E4:67_1_2_3"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:67_1_2_3"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E6:67_1_2_3"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:67_1_2_3"/>
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</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><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">hence </span><a NAME="E7:67"/>
 <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>,<a href="borsuk_6.html#K5" title="BORSUK_6:func.5">3RP</a>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c8">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</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>
