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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">f2</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">X</font>, <font color="Olive" title="b4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b5">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b5">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b4">Y</font> holds <br/><font color="Olive" title="b5">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Olive" title="b2">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b4">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">f2</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">X</font>, <font color="Olive" title="b3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b4">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Olive" title="b1">f2</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/><font color="Olive" title="b4">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Olive" title="b1">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f2</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">X</font>, <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b3">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">f2</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/><font color="Olive" title="b3">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Olive" title="b1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">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">  for <font color="Olive" title="b1">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c5">Y</font>
 ; <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">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">r1</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="E4:51"/><span class="lab"><font color="Green" title="E22">A3</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">X</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="c6">f1</font></span>)</span> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="cfunct_1.html#T69" title="CFUNCT_1:th.69">CFUNCT_1:69</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:51"/><span class="lab"><font color="Green" title="E23">A4</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="c5">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">f2</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">f2</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="c8">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="vfunct_2.html#D3" target="_self" title="VFUNCT_2:def.3">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">r1</font> = <font color="Maroon" title="c7">r1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c9">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r2</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">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font></span>)</span> <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">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <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="c10">r</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>; <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="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font></span>)</span> <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">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">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="c10">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</font></span>)</span> <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">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span></span>)</span>
 ; <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#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">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="c10">r</font></span><br/><a NAME="E2:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">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="E3:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="E4:51_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E24">A5</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="E5:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <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">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def3</a></span>;<br/><a NAME="E6:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="c6">f1</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="E7:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c6">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E26">A7</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="E8:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/><a NAME="E9:51_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E25">A6</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="E10:51_1"/>
<font color="Maroon" title="c11">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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E28">A9</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="E11:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_1"><span class="lab"><font color="Green" title="E30">A11</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="E12:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A12</font></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">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">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="c8">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/><a NAME="E13:51_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T105" title="CLVECT_1:th.105">CLVECT_1:105</a>, <a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/><a NAME="E14:51_1"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">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="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E12:51_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E15:51_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"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">c</font></span>)</span></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="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D13" title="CLVECT_1:def.13">CLVECT_1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E16:51_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">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="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="vfunct_2.html#D1" target="_self" title="VFUNCT_2:def.1">Def3</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="E9:51"/>
<font color="Maroon" title="c6">f1</font> <a href="vfunct_2.html#K1" title="VFUNCT_2:func.1">(#)</a> <font color="Maroon" title="c3">f2</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">Y</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>; <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>
