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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>thus </b><a NAME="E1:90"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a> &amp; <font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a> implies <font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V1">bounded_above</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:90_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:90_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 ;<br/>

<a NAME="E3:90_1"/>
 <a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T9" target="_self">Th9</a>, <a class="ref" href="seqm_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:90_1"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, <a class="ref" href="seqm_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:90_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:90"/>
( <font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a> &amp; <font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a> implies <font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V2">bounded_below</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:90_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:90_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 ;<br/>

<a NAME="E3:90_2"/>
 <a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T9" target="_self">Th9</a>, <a class="ref" href="seqm_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:90_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font></span>)</span> is <a href="seq_2.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, <a class="ref" href="seqm_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:90_2"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E3:90"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>and </b><br/><a NAME="E4:90"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 ;<br/>

<a NAME="E5:90"/>
 <a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T9" target="_self">Th9</a>, <a class="ref" href="seqm_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:90"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">seq1</font></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, <a class="ref" href="seqm_3.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E7:90"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>

<a NAME="E8:90"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E9:90"/>
<font color="Maroon">seq1</font> <a href="seq_1.html#K10">-</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, </i>;<br/>

<b>thus </b><a NAME="E10:90"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K11">(#)</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T7" target="_self">Th7</a>, <a class="ref" href="seqm_3.html#T10" target="_self">Th10</a>, </i>;<br/>


</div>
