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<b>let </b><font color="Maroon">f</font> be   <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 <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">g</font> be   <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 <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Olive">p</font>,<font color="Olive">q</font>
 <b>and </b><br/><a NAME="E2:32_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Olive">p</font>,<font color="Olive">q</font>
 ;<br/>

<a NAME="E3:32_1_2"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">p</font>,<font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>



<b>thus </b><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> = 
 <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_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="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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