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<div class="add">

<b>let </b><font color="Maroon" title="c22">C1</font>, <font color="Maroon" title="c23">C2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>, <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green" title="E66">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> holds  <span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green" title="E67">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> holds  <span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">C2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
 ;<br/>

<a NAME="E3:47_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> holds <br/><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1_2_1"/><i><font color="Green" title="E68">A6</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>
 ;<br/>

<a NAME="E2:47_1_2_1"/><b>then </b><i><font color="Green" title="E69">A7</font></i>: 
<font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:47_1_2_1"/><i><font color="Green" title="E70">A8</font></i>: 
<font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="realset2.html#T6" target="_self" title="REALSET2:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E2:47_1_2"><i><font color="Green" title="E67">A5</font></i></a>, <a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green" title="E68">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>,<font color="Maroon" title="c24">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green" title="E68">A6</font></i></a>, <a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="txt" href="realset2.html#E3:47_1_2_1"><i><font color="Green" title="E70">A8</font></i></a>, <a class="ref" href="realset2.html#T4" target="_self" title="REALSET2:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green" title="E68">A6</font></i></a>, <a class="txt" href="realset2.html#E2:47_1_2_1"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="realset2.html#T5" target="_self" title="REALSET2:th.5">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<b>by </b><i><a class="txt" href="realset2.html#E1:47_1_2"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="realset2.html#E1:47_1_2_1"><i><font color="Green" title="E68">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<b>by </b><i><a class="txt" href="realset2.html#E3:47_1_2_1"><i><font color="Green" title="E70">A8</font></i></a>, <a class="ref" href="realset2.html#T6" target="_self" title="REALSET2:th.6">Th6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_2"/>
<font color="Maroon" title="c22">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">C2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


</div>
