let s1, s2 be State of SCMPDS ; :: thesis: ( s1,s2 equal_outside NAT implies for a being Int_position holds s1 . a = s2 . a )
set IL = NAT ;
assume A1: s1,s2 equal_outside NAT ; :: thesis: for a being Int_position holds s1 . a = s2 . a
let a be Int_position ; :: thesis: s1 . a = s2 . a
A2: a in dom s1 by SCMPDS_2:49;
A3: a in dom s2 by SCMPDS_2:49;
a in SCM-Data-Loc by SCMPDS_2:def 2;
then A4: not a in NAT by AMI_2:29, XBOOLE_0:3;
then a in (dom s1) \ NAT by A2, XBOOLE_0:def 5;
then A5: a in (dom s1) /\ ((dom s1) \ NAT ) by XBOOLE_0:def 4;
a in (dom s2) \ NAT by A3, A4, XBOOLE_0:def 5;
then A6: a in (dom s2) /\ ((dom s2) \ NAT ) by XBOOLE_0:def 4;
thus s1 . a = (s1 | ((dom s1) \ NAT )) . a by A5, FUNCT_1:71
.= (s2 | ((dom s2) \ NAT )) . a by A1, FUNCT_7:def 2
.= s2 . a by A6, FUNCT_1:71 ; :: thesis: verum