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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b3">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">h1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Olive" title="b2">h2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b2">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b3">seq</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c3">h1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c4">h2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">seq</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c5">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">h1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c4">h2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> ) )</span><br/>







<b>assume </b><a NAME="E1:39"/>
( <font color="Maroon" title="c3">h1</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c4">h2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> )</span><br/>

<a NAME="E2:39"/><b>then </b>
<font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#T38" title="VFUNCT_1:th.38">VFUNCT_1:38</a></i>;<br/>
<a NAME="E3:39"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/>
<a NAME="E5:39"/><b>then </b><i><font color="Green" title="E29">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E6:39"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span></span><br/>

<b>thus </b><a NAME="E7:39"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">seq</font></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E5:39"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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