theorem Th5:
  s.a = (s +* Start-At(loc,SCM R)).a
proof
  a in the carrier of SCM R;
  then a in dom s by PARTFUN1:def 2;
  then
A1: dom (Start-At(loc,SCM R)) = {IC SCM R} & a in dom s \/
dom (Start-At(loc,SCM R)) by XBOOLE_0:def 3;
  a <> IC SCM R by SCMRING3:2;
  then not a in {IC SCM R} by TARSKI:def 1;
  hence thesis by A1,FUNCT_4:def 1;
end;
