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: now :: thesis: ( ( for a being Int_position holds s1 . a = s2 . a ) implies for x being set st x in SCM-Data-Loc holds
s1 . x = s2 . x )

assume A2: 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 AMI_2:def 16;
hence s1 . x = s2 . x by A2; :: thesis: verum

end;

end;

A3: now :: thesis: ( ( for x being set st x in SCM-Data-Loc holds
s1 . x = s2 . x ) implies for a being Int_position holds s1 . a = s2 . a )

assume A4: 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 AMI_2:def 16;
hence s1 . a = s2 . a by A4; :: thesis: verum

end;

end;

dom s2 = the carrier of SCMPDS by PARTFUN1:def 2;
then dom s2 = {(IC )} \/ SCM-Data-Loc by SCMPDS_2:84, STRUCT_0:4;
then A5: SCM-Data-Loc c= dom s2 by XBOOLE_1:7;
dom s1 = the carrier of SCMPDS by PARTFUN1:def 2;
then dom s1 = {(IC )} \/ SCM-Data-Loc by SCMPDS_2:84, STRUCT_0:4;
then SCM-Data-Loc c= dom s1 by XBOOLE_1:7;
hence ( ( for a being Int_position holds s1 . a = s2 . a ) iff DataPart s1 = DataPart s2 ) by A5, A1, A3, FUNCT_1:95, SCMPDS_2:84; :: thesis: verum