reserve x,y for set;
reserve i, j, k for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM;
reserve a, b for Data-Location,
  loc for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM,
  da,db for Data-Location;

theorem
  for s being State of SCM, iloc being Nat, a
  being Data-Location holds s.a = (s +* Start-At(iloc,SCM)).a
proof
  let s be State of SCM, iloc be Nat, a be
  Data-Location;
  a in the carrier of SCM;
  then a in dom s by PARTFUN1:def 2;
  then
A1: dom (Start-At(iloc,SCM)) = {IC SCM} &
 a in dom s \/ dom (Start-At(iloc,SCM)) by XBOOLE_0:def 3;
  a <> IC SCM by Th2;
  then not a in {IC SCM} by TARSKI:def 1;
  hence thesis by A1,FUNCT_4:def 1;
end;
