<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  implies <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:96_1_1_1"/>
( <font color="Maroon" title="c1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  implies <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c2">z</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><span class="kw">assume </span><a NAME="E3:96_1_1_1_1"/>
<font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> </span><br/><span class="kw">thus </span><a NAME="E4:96_1_1_1_1"/>
<font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D8" title="SEQ_2:def.8">SEQ_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">z</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D1" title="SEQ_2:def.1">SEQ_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Olive">b<sub>1</sub></font>) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">x</font>) )</span><br/>

<span class="kw">thus </span><a NAME="E1:96_1_1_1_1_1"/>
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">x</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">take </span>
<font color="Maroon" title="c2">z</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D2" title="SEQ_2:def.2">SEQ_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Olive">b<sub>1</sub></font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> )</span><br/><span class="kw">thus </span><a NAME="E5:96_1_1_1_1"/>
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
not <font color="Maroon" title="c1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  implies <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:96_1_1_1_2"/>
<font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:96_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1_1_2"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="valued_0.html#T15" title="VALUED_0:th.15">VALUED_0:15</a></span>;<br/><span class="kw">thus </span><a NAME="E5:96_1_1_1_2"/>
<font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D8" title="SEQ_2:def.8">SEQ_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D1" title="SEQ_2:def.1">SEQ_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Olive">b<sub>1</sub></font>) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1_1_1_2_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>)</span><br/>

<a NAME="E2:96_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96_1_1_1_2_1"/>
not <font color="Maroon" title="c2">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">take </span>
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D2" title="SEQ_2:def.2">SEQ_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Olive">b<sub>1</sub></font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c1">f</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E6:96_1_1_1_2"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span><br/><a NAME="E7:96_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:96_1_1_1_2"/>
not <a href="seq_1.html#K1" title="SEQ_1:func.1">K247</a>(<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
