<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" 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> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b5">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> &amp; (  for <font color="Olive" title="b6">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b6">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b6">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" 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>, <font color="Olive" title="b2">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">s</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="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b4">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c6">F</font> =  <a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>; <a class="txt" 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="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</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="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <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="c3">b</font> ) ) )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D7" target="_self" title="TOPALG_1:def.7">Def7</a>, <a class="ref" href="borsuk_1.html#D14" title="BORSUK_1:def.14">BORSUK_1:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>
; <a class="txt" 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="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</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="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <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="c3">b</font> ) ) )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">s</font>,1) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D7" target="_self" title="TOPALG_1:def.7">Def7</a>, <a class="ref" href="borsuk_1.html#D15" title="BORSUK_1:def.15">BORSUK_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>
; <a class="txt" 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="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> holds <br/> ( <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <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="c3">b</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>; <a class="txt" 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="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c8">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c8">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<a NAME="E3:78"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
( <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c8">t</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D7" target="_self" title="TOPALG_1:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
; <a class="txt" 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="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c8">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E6:78"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
( <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="topalg_1.html#K7" title="TOPALG_1:func.7">RealHomotopy</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c8">t</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D7" target="_self" title="TOPALG_1:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T29" title="EUCLID:th.29">EUCLID:29</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
