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<span class="kw">let </span><font color="Maroon" title="c1">L1</font> be   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LinearFunc</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">  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">L1</font> is   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LinearFunc</a></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">g1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D3" onmouseover="rs('fdiff_1/D3')" onmouseout="rh()">FDIFF_1:def 3</a></span>;<br/>
<a NAME="E3:24"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c1">L1</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D3" onmouseover="rs('fdiff_1/D3')" onmouseout="rh()">FDIFF_1:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_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">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">p</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">p</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"> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">pp</font> = <font color="Maroon" title="c3">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">L1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">p</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T58" onmouseover="rs('rfunct_1/T58')" onmouseout="rh()">RFUNCT_1:58</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">g1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:24"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c1">L1</font> is   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LinearFunc</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D3" onmouseover="rs('fdiff_1/D3')" onmouseout="rh()">FDIFF_1:def 3</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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