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: ( ( 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 ) by AMI_2:def 16;
A4: dom s2 = the carrier of SCMPDS by PARTFUN1:def 2;
dom s1 = the carrier of SCMPDS by PARTFUN1:def 2;
hence ( ( for a being Int_position holds s1 . a = s2 . a ) iff DataPart s1 = DataPart s2 ) by A4, A1, A3, FUNCT_1:95, SCMPDS_2:84; :: thesis: verum