let s be SCMPDS-State; :: thesis: for u, v being Nat holds (SCM-Chg s,u) . v = s . v
let u, v be Nat; :: thesis: (SCM-Chg s,u) . v = s . v
v in NAT by ORDINAL1:def 13;
then A1: ( SCMPDS-OK . NAT = NAT & SCMPDS-OK . v = SCMPDS-Instr ) by Th18, Th20, AMI_2:30;
{NAT } = dom (NAT .--> u) by FUNCOP_1:19;
then not v in dom (NAT .--> u) by A1, Th17, TARSKI:def 1;
hence (SCM-Chg s,u) . v = s . v by FUNCT_4:12; :: thesis: verum