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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b3">Y</font> holds <br/>( <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Olive" title="b2">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b3">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b2">Y</font> holds <br/>( <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Olive" title="b1">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b1">Y</font> holds <br/>( <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b1">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font> implies ( <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp;  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">c</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_2.html#D3" target="_self" title="VFUNCT_2:def.3">Def7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">c</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font></span><br/><a NAME="E2:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:49_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D3" title="NORMSP_0:def.3">NORMSP_0:def 3</a></span>;<br/><a NAME="E5:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:49_1"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="clvect_1.html#T105" title="CLVECT_1:th.105">CLVECT_1:105</a></span>;<br/><a NAME="E7:49_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:49_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="normsp_0.html#D3" title="NORMSP_0:def.3">NORMSP_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:49"/>
<span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">RFUNCT_1:73</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E6:49"/>
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="vfunct_2.html#K2" title="VFUNCT_2:func.2">(#)</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="vfunct_2.html#T44" target="_self" title="VFUNCT_2:th.44">Th44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:49"/>
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_2.html#T23" target="_self" title="VFUNCT_2:th.23">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
