let s be SCM-State; :: thesis: for u being Nat
for mk being Element of SCM-Data-Loc holds (SCM-Chg s,u) . mk = s . mk
let u be Nat; :: thesis: for mk being Element of SCM-Data-Loc holds (SCM-Chg s,u) . mk = s . mk
let mk be Element of SCM-Data-Loc ; :: thesis: (SCM-Chg s,u) . mk = s . mk
A1: ( SCM-OK . NAT = NAT & SCM-OK . mk = INT ) by Lm4, Th7, Th8;
{NAT } = dom (NAT .--> u) by FUNCOP_1:19;
then not mk in dom (NAT .--> u) by A1, NUMBERS:27, TARSKI:def 1;
hence (SCM-Chg s,u) . mk = s . mk by FUNCT_4:12; :: thesis: verum