set T1 = {(IC )};
set T2 = SCM-Data-Loc ;
set T3 = NAT ;
let s1, s2 be State of SCMPDS; :: thesis: ( ( for a being Int_position holds s1 . a = s2 . a ) iff DataPart s1 = DataPart s2 )
A1: ({(IC )} \/ SCM-Data-Loc) \/ NAT = SCM-Data-Loc \/ ({(IC )} \/ NAT) by XBOOLE_1:4;

A2: now

assume A3: for a being Int_position holds s1 . a = s2 . a ; :: thesis: for x being set st x in SCM-Data-Loc holds
s1 . x = s2 . x

hereby :: thesis: verum

let x be set ; :: thesis: ( x in SCM-Data-Loc implies s1 . x = s2 . x )
assume x in SCM-Data-Loc ; :: thesis: s1 . x = s2 . x
then x is Int_position by SCMPDS_2:9;
hence s1 . x = s2 . x by A3; :: thesis: verum

end;

end;

A4: now

assume A5: for x being set st x in SCM-Data-Loc holds
s1 . x = s2 . x ; :: thesis: for a being Int_position holds s1 . a = s2 . a

hereby :: thesis: verum

let a be Int_position ; :: thesis: s1 . a = s2 . a
a in SCM-Data-Loc by SCMPDS_2:def 2;
hence s1 . a = s2 . a by A5; :: thesis: verum

end;

end;

dom s2 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom s2 = ({(IC )} \/ SCM-Data-Loc) \/ NAT by COMPOS_1:160, AMI_3:4, SCMPDS_2:100;
then A6: SCM-Data-Loc c= dom s2 by A1, XBOOLE_1:7;
dom s1 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom s1 = ({(IC )} \/ SCM-Data-Loc) \/ NAT by COMPOS_1:160, AMI_3:4, SCMPDS_2:100;
then SCM-Data-Loc c= dom s1 by A1, XBOOLE_1:7;
hence ( ( for a being Int_position holds s1 . a = s2 . a ) iff DataPart s1 = DataPart s2 ) by A6, A2, A4, FUNCT_1:165, SCMPDS_2:100; :: thesis: verum