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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <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">f</font>, <font color="Olive" title="b2">h</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K9" title="CLOPBAN1:func.9">C_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span><br/> for <font color="Olive" title="b3">c</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds <br/> ( <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">f</font> iff for <font color="Olive" title="b4">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b2">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b4">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">h</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K9" title="CLOPBAN1:func.9">C_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <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">c</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds <br/> ( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> iff for <font color="Olive" title="b2">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b2">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">c</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> iff for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>





<a NAME="E1:51"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="clopban1.html#K9" title="CLOPBAN1:func.9">C_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of  <a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#T24" target="_self" title="CLOPBAN1:th.24">Th24</a>, <a class="ref" href="csspace.html#T13" title="CSSPACE:th.13">CSSPACE:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c3">f</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c4">h</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></span>;<br/>
<div><span class="kw">hereby </span> <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">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 ; <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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.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"> <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><br/><a NAME="E2:51_1"/>
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="clvect_1.html#T34" title="CLVECT_1:th.34">CLVECT_1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1"/>
<font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#T19" target="_self" title="CLOPBAN1:th.19">Th19</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">assume </span><a NAME="E5:51"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">h</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K10" title="CLOPBAN1:func.10">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E6:51"/><span class="kw">then </span>
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban1.html#T19" target="_self" title="CLOPBAN1:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="clvect_1.html#T34" title="CLVECT_1:th.34">CLVECT_1:34</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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