<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">X0</font>, <font color="Olive" title="b2">X1</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">X0</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b4">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Olive" title="b1">X0</font> &amp; <font color="Olive" title="b2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X0</font> holds <br/><font color="Olive" title="b4">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Olive" title="b2">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Olive" title="b2">X1</font></span><br/><b>let </b><font color="Maroon" title="c3">X0</font>, <font color="Maroon" title="c4">X1</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</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">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X0</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b2">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c4">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X0</font> holds <br/><font color="Olive" title="b2">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font></span><br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</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">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X0</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c4">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X0</font> holds <br/><font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font></span><br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X0</font>,<font color="Maroon" title="c2">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="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X0</font> &amp; <font color="Maroon" title="c4">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X0</font> implies <font color="Maroon" title="c6">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font> )</span><br/>











<b>assume </b><a NAME="E1:74"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X0</font> or <font color="Maroon" title="c6">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font> )</span><br/>

<b>assume </b><a NAME="E2:74"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c4">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font></span><br/>

<a NAME="E3:74"/><b>then </b><i><font color="Green" title="E56">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X0</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X1</font> = 
<font color="Maroon" title="c6">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X1</font>
<b>by </b><i><a class="txt" href="#E2:74"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="tmap_1.html#D4" target="_self" title="TMAP_1:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X1</font>
<b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E3:74"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</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>
