let s be SCM+FSA-State; :: thesis: for u being Element of NAT
for mk being Element of SCM+FSA-Data-Loc holds (SCM+FSA-Chg (s,u)) . mk = s . mk
let u be Element of NAT ; :: thesis: for mk being Element of SCM+FSA-Data-Loc holds (SCM+FSA-Chg (s,u)) . mk = s . mk
let mk be Element of SCM+FSA-Data-Loc ; :: thesis: (SCM+FSA-Chg (s,u)) . mk = s . mk
( SCM+FSA-OK . mk = INT & {NAT} = dom (NAT .--> u) ) by Th10, FUNCOP_1:13;
then not mk in dom (NAT .--> u) by Th9, NUMBERS:27, TARSKI:def 1;
hence (SCM+FSA-Chg (s,u)) . mk = s . mk by FUNCT_4:11; :: thesis: verum