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 . v = SCMPDS-Instr by Th20;
( SCMPDS-OK . NAT = NAT & {NAT} = dom (NAT .--> u) ) by Th18, AMI_2:30, 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