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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Z</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/>



<a NAME="E1:19"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span> #) &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Z</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Z</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Z</font></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="waybel25.html#D2">WAYBEL25:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">Z</font></span>)</span> <b>by </b><i><a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<b>set </b><font color="Maroon">Xf</font> =  <a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">a</font>, <font color="Maroon">g2</font> = <font color="Maroon">b</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:19"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E5:19"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">g1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:19"/>
( <font color="Maroon">g1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">g2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font> &amp; <span class="p1">(<span class="default"><a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">fg1</font> = <font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font>, <font color="Maroon">fg2</font> = <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/>
<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">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E3:19_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E4:19_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:19"/><b>then </b>
<font color="Maroon">fg1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">fg2</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:19"/>
<span class="p1">(<span class="default"><a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K4" target="_self">oContMaps</a> <font color="Maroon">f</font>,<font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


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