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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></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="Maroon">a<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Olive">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Olive">b1</font></span>)</span> );<br/>
<a NAME="E1:6_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>   ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:6_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
;<br/>
<a NAME="E2:6_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">b1</font> = <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span>;<br/>
<a NAME="E4:6_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:6_1_1_1"/><i><font color="Green">E43</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E6:6_1_1_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<a NAME="E7:6_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:6_1_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is  <a href="jordan1.html#V1">convex</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:6_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E10:6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b1</font></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:6_1_1_1"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Olive">z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <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="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E5:6_1_1"/>
 ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</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="Maroon">y</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Olive">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <font color="Olive">b1</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
for <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</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="Maroon">y</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Olive">a1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K18">*</a> <font color="Olive">b1</font></span>)</span>
 ;<br/>


</div>
