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<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">E</font>;<br/>















<b>assume </b><a NAME="E1:47"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d2</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funcop_1.html#K6">.:</a> <font color="Maroon">f</font>,<font color="Maroon">f'</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funcop_1.html#K6">.:</a> <font color="Maroon">f</font>,<font color="Maroon">f'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T48">FUNCOP_1:48</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T48">FUNCOP_1:48</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funcop_1.html#K6">.:</a> <font color="Maroon">f</font>,<font color="Maroon">f'</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">H</font> <a href="funcop_1.html#K6">.:</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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