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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">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K5" title="ORDEQ_01:func.5">R_NormSpace_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="b3">g</font> being   <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="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/> for <font color="Olive" title="b4">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></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">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K5" title="ORDEQ_01:func.5">R_NormSpace_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="b2">g</font> being   <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="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> holds <br/> for <font color="Olive" title="b3">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="ordeq_01.html#K5" title="ORDEQ_01:func.5">R_NormSpace_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">g</font> being   <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">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> holds <br/> for <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <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="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> implies  for <font color="Olive" title="b1">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</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">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">f</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><span class="kw"> such that </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordeq_01.html#D2" target="_self" onmouseover="rs('ordeq_01/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">g1</font> = <font color="Maroon" title="c4">g</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="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="ordeq_01.html#T9" target="_self" onmouseover="rs('ordeq_01/T9')" onmouseout="rh()">Th9</a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c3">v</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#K9" title="RSSPACE4:func.9">R_NormSpace_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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:42"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">v1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">t</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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"> ( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E7:42"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">t1</font> = <font color="Maroon" title="c8">t</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E9:42"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c7">v1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="rsspace4.html#T16" onmouseover="rs('rsspace4/T16')" onmouseout="rh()">RSSPACE4:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:42"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E27">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/>


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