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
A1: ( SCM+FSA-OK . NAT = NAT & SCM+FSA-OK . mk = INT ) by Th9, Th10;
{NAT } = dom (NAT .--> u) by FUNCOP_1:19;
then not mk in dom (NAT .--> u) by A1, Th13, TARSKI:def 1;
hence (SCM+FSA-Chg s,u) . mk = s . mk by FUNCT_4:12; :: thesis: verum