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<span class="kw">let </span><font color="Maroon" title="c1">X</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y</font> holds <br/> ( <font color="Olive" title="b2">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  iff  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Olive" title="b2">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  iff  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Olive" title="b1">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>; <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="c3">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  iff  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">K</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">assume </span><a NAME="E2:38_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">K</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon" title="c3">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> </span><br/><a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">r0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">r0</font> = <font color="Maroon" title="c5">r0</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="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font></span><br/><a NAME="E2:38_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 ;<br/><span class="kw">hence </span><a NAME="E3:38_1_1_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> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">CLVECT_1:106</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>
<a NAME="E5:38_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38_1_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> <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</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="c5">K</font> = <font color="Maroon" title="c4">K</font>; <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="c3">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">K</font></span><br/><a NAME="E1:38_1_2"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font>
 ;<br/><span class="kw">hence </span><a NAME="E2:38_1_2"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</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">hence </span><a NAME="E5:38_1"/>
<font color="Maroon" title="c3">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="csspace4.html#D4" target="_self" title="CSSPACE4:def.4">Def4</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">hence </span><a NAME="E2:38"/>
( <font color="Maroon" title="c3">g</font> is  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  iff  <a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">g</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#T14" target="_self" title="CSSPACE4:th.14">Th14</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/>


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