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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="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K2" title="ORDEQ_01:func.2">R_VectorSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font>)</span>)</span><br/>  for <font color="Olive" title="b5">f9</font>, <font color="Olive" title="b6">g9</font>, <font color="Olive" title="b7">h9</font> being   <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Olive" title="b1">Y</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b6">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> &amp; <font color="Olive" title="b7">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b6">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">g</font> iff  for <font color="Olive" title="b8">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b7">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b8">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b8">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K2" title="ORDEQ_01:func.2">R_VectorSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span><br/>  for <font color="Olive" title="b4">f9</font>, <font color="Olive" title="b5">g9</font>, <font color="Olive" title="b6">h9</font> being   <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b5">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b6">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b5">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b6">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">g</font> iff  for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b6">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K2" title="ORDEQ_01:func.2">R_VectorSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f9</font>, <font color="Olive" title="b2">g9</font>, <font color="Olive" title="b3">h9</font> being   <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Olive" title="b3">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/>( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> iff  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b3">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font></span>)</span> )</span><br/>





<a NAME="E1:19"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="ordeq_01.html#K2" title="ORDEQ_01:func.2">R_VectorSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>) is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</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="rsspace.html#T11" onmouseover="rs('rsspace/T11')" onmouseout="rh()">RSSPACE:11</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="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</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="rlsub_1.html#T10" onmouseover="rs('rlsub_1/T10')" onmouseout="rh()">RLSUB_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c5">h</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</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="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="rlsub_1.html#T10" onmouseover="rs('rlsub_1/T10')" onmouseout="rh()">RLSUB_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c4">g</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</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="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="rlsub_1.html#T10" onmouseover="rs('rlsub_1/T10')" onmouseout="rh()">RLSUB_1:10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">f9</font>, <font color="Maroon" title="c10">g9</font>, <font color="Maroon" title="c11">h9</font> be   <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c10">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c11">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> iff  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E5:19"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c9">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c10">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c11">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> iff  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">f90</font> = <font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c13">g90</font> = <font color="Maroon" title="c8">g1</font>, <font color="Maroon" title="c14">h90</font> = <font color="Maroon" title="c7">h1</font> as   <a href="rsspace4.html#V1" title="RSSPACE4:attr.1">bounded</a>  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <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="rsspace4.html#D5" onmouseover="rs('rsspace4/D5')" onmouseout="rh()">RSSPACE4:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E9">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/><a NAME="E2:19_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="rlsub_1.html#T13" onmouseover="rs('rlsub_1/T13')" onmouseout="rh()">RLSUB_1:13</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> = 
<font color="Maroon" title="c14">h90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f90</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g90</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="rsspace4.html#T8" onmouseover="rs('rsspace4/T8')" onmouseout="rh()">RSSPACE4:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c14">h90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">h90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c14">h90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> = 
<font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f90</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g90</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g90</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E3:19_2"/><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="c6">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace4.html#T8" onmouseover="rs('rsspace4/T8')" onmouseout="rh()">RSSPACE4:8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_2"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="rlsub_1.html#T13" onmouseover="rs('rlsub_1/T13')" onmouseout="rh()">RLSUB_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:19"/>
( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> iff  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c11">h9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
