reserve l, m, n for Nat;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;
reserve L for finite Subset of Int-Locations;
reserve L for finite Subset of FinSeq-Locations;
reserve L for finite Subset of Int-Locations;

theorem Th27:
  for Iloc being Subset of Int-Locations, Floc being Subset of
  FinSeq-Locations holds s1 | (Iloc \/ Floc) = s2 | (Iloc \/ Floc) iff (for x
  being Int-Location st x in Iloc holds s1.x = s2.x) & for x being
  FinSeq-Location st x in Floc holds s1.x = s2.x
proof
  let Iloc be Subset of Int-Locations, Floc be Subset of FinSeq-Locations;
  FinSeq-Locations c= dom s1 by SCMFSA_2:46;
  then
A1: Floc c= dom s1;
  FinSeq-Locations c= dom s2 by SCMFSA_2:46;
  then
A2: Floc c= dom s2;
  Int-Locations c= dom s2 by SCMFSA_2:45;
  then
A3: Iloc c= dom s2;
  then
A4: Iloc \/ Floc c= dom s2 by A2,XBOOLE_1:8;
  Int-Locations c= dom s1 by SCMFSA_2:45;
  then
A5: Iloc c= dom s1;
  then
A6: Iloc \/ Floc c= dom s1 by A1,XBOOLE_1:8;
  hereby
    assume
A7: s1 | (Iloc \/ Floc) = s2 | (Iloc \/ Floc);
    hereby
      let x be Int-Location;
      assume x in Iloc;
      then x in Iloc \/ Floc by XBOOLE_0:def 3;
      hence s1.x = s2.x by A6,A4,A7,FUNCT_1:95;
    end;
    let x be FinSeq-Location;
    assume x in Floc;
    then x in Iloc \/ Floc by XBOOLE_0:def 3;
    hence s1.x = s2.x by A6,A4,A7,FUNCT_1:95;
  end;
  assume that
A8: for x being Int-Location st x in Iloc holds s1.x = s2.x and
A9: for x being FinSeq-Location st x in Floc holds s1.x = s2.x;
A10: now
    hereby
      let x be set;
      assume
A11:  x in Iloc;
      then x in Int-Locations;
      then reconsider x9 = x as Int-Location by AMI_2:def 16;
      thus s1.x = s2.x9 by A8,A11
        .= s2.x;
    end;
    let x be set;
    assume
A12: x in Floc;
    then x in FinSeq-Locations;
    then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
    thus s1.x = s2.x9 by A9,A12
      .= s2.x;
  end;
  then
A13: s1 | Floc = s2 | Floc by A1,A2,FUNCT_1:95;
  s1 | Iloc = s2 | Iloc by A5,A3,A10,FUNCT_1:95;
  hence thesis by A13,RELAT_1:150;
end;
