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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>5</sub></font>() = <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span>);<br/>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>,<span class="p2"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">F<sub>5</sub></font>()<b> such that </b><br/><a NAME="E2:16_1"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds  <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c1">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">y1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>let </b><font color="Maroon" title="c6">x3</font>, <font color="Maroon" title="c7">y3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>();<br/><b>let </b><font color="Maroon" title="c8">x4</font>, <font color="Maroon" title="c9">y4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>4</sub></font>();<br/><b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>;<br/>









<b>assume </b><a NAME="E3:16_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c8">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c9">y4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> )
 ;<br/>

<a NAME="E4:16_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T60" title="MCART_1:th.60">MCART_1:60</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  &amp; <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  &amp; <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  &amp; <font color="Maroon" title="c6">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  &amp; <font color="Maroon" title="c7">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  &amp; <font color="Maroon" title="c8">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  &amp; <font color="Maroon" title="c9">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  )
 
<b>by </b><i><a class="txt" href="metric_3.html#E3:16_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>)
 <b>by </b><i><a class="txt" href="metric_3.html#E2:16_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


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