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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>







<i><font color="Green">E37</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
;<br/>
<b>assume </b><a NAME="E2:28"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">g</font>,<font color="Maroon">X</font> <a href="tietze.html#R1" target="_self">is_absolutely_bounded_by</a> <font color="Maroon">r</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tietze.html#D1" target="_self">TIETZE:def 1</a><br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:28"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:28"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:28"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, , </i>;<br/>
<a NAME="E7:28"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E8:28"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E9:28"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K4">-</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a>, </i>;<br/>


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