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<b>consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:71_1_1"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c5">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<a NAME="E3:71_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">vx</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c7">wx</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E5:71_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">vx</font>,<font color="Maroon" title="c7">wx</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E2:71_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">vy</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c9">wy</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E7:71_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">vy</font>,<font color="Maroon" title="c9">wy</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E2:71_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">vx</font>,<font color="Maroon" title="c7">wx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">vy</font>,<font color="Maroon" title="c9">wy</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:71_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">vx</font>,<font color="Maroon" title="c7">wx</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">vy</font>,<font color="Maroon" title="c9">wy</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self" title="HERMITAN:def.8">Def8</a></i>;<br/><a NAME="E3:71_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">vx</font>,<font color="Maroon" title="c7">wx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">vy</font>,<font color="Maroon" title="c9">wy</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self" title="HERMITAN:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E4:71_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:71_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E5:71_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E7:71_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:71_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>) &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="#E2:71_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E3:71_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E5:71_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E7:71_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</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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