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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds  <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">t</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:74_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds  <font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">t</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>
<a NAME="E3:74_1_2"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E4:74_1_2"/>
for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds  <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,<font color="Olive">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> , <font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">t</font>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:74_1_2"><i><font color="Green">E46</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:74_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E2:74_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
