let i be Element of SCM+FSA-Memory ; :: thesis: ( SCM+FSA-OK . i = NAT implies i = NAT )
assume A1: SCM+FSA-OK . i = NAT ; :: thesis: i = NAT
not i in NAT by A1, Th11, Th13;
then A2: i in ({NAT } \/ SCM+FSA-Data-Loc ) \/ SCM+FSA-Data*-Loc by Th8, XBOOLE_0:def 3;
not i in SCM+FSA-Data*-Loc by A1, Th12, Th13;
then A3: i in {NAT } \/ SCM+FSA-Data-Loc by A2, XBOOLE_0:def 3;
not i in SCM+FSA-Data-Loc by A1, Th10, Th13, NUMBERS:27;
then i in {NAT } by A3, XBOOLE_0:def 3;
hence i = NAT by TARSKI:def 1; :: thesis: verum