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<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <font color="Maroon">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <font color="Maroon">Z</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="lopban_1.html#V3">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="lopban_1.html#V3">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/>





<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:3_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E2:3_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T38">LOPBAN_1:38</a></i>;<br/><a NAME="E3:3_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T38">LOPBAN_1:38</a></i>;<br/><a NAME="E4:3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E5:3_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E6:3_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E5:3"/><b>then </b><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>
<b>set </b><font color="Maroon">K</font> = <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span>;<br/>
<a NAME="E6:3"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/>
<a NAME="E7:3"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:3_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:3_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:3_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as  <a href="lopban_1.html#V3">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font> <b>by </b><i>, , , <a class="ref" href="lopban_1.html#D9">LOPBAN_1:def 9</a></i>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:3_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14">PreNorms</a> <font color="Maroon">gf</font>
 ;<br/><a NAME="E2:3_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">gf</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> where <font color="Olive">t</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:3_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gf</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">t</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E5:3_3"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E11:3"/>
( ( for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="lopban_1.html#K14">PreNorms</a> <font color="Maroon">gf</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span> ) implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14">PreNorms</a> <font color="Maroon">gf</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">gg</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">ff</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>
<b>hence </b><a NAME="E12:3"/>
( <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is  <a href="lopban_1.html#V3">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font> &amp; ( for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds <br/> ( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">f</font></span>)</span> ) ) )
 <b>by </b><i>, , , , <a class="ref" href="lopban_1.html#T36">LOPBAN_1:36</a></i>;<br/>


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