reserve x,y,z for set,
  k for Nat;
reserve J,J1,K for Element of Segm 13,
  a for Nat,
  b,b1,b2,c,c1,c2 for Element of SCM+FSA-Data-Loc,
  f,f1,f2 for Element of SCM+FSA-Data*-Loc;

theorem Th8:
  for s being SCM+FSA-State, s1 being SCM-State
   holds s +* s1 is SCM+FSA-State
proof
  let s be SCM+FSA-State, s1 be SCM-State;
A1: dom(SCM*-VAL*SCM+FSA-OK) = SCM+FSA-Memory by Lm8;
  then reconsider
  f = (SCM*-VAL*SCM+FSA-OK) as non-empty ManySortedSet of SCM+FSA-Memory
            by PARTFUN1:def 2;
A2: dom s1 = dom(SCM-VAL*SCM-OK) by CARD_3:9
    .= SCM-Memory by AMI_2:27;
  now
    let x be set;
    assume
A3: x in dom s1;
    then
A4: x in {NAT} \/ SCM-Data-Loc by A2;
    per cases by A4,XBOOLE_0:def 3;
    suppose
A5:   x in {NAT};
      reconsider a = x as Element of SCM-Memory by A2,A3;
A6:   s1.a in pi(product(SCM-VAL*SCM-OK),a) by CARD_3:def 6;
A7:   x = NAT by A5,TARSKI:def 1;
      dom(SCM-VAL*SCM-OK) = SCM-Memory by AMI_2:27;
      then pi(product(SCM-VAL*SCM-OK),a) = (SCM-VAL*SCM-OK).a by CARD_3:12
        .= NAT by A7,AMI_2:6;
      hence s1.x in f.x by A5,A6,Th4,TARSKI:def 1;
    end;
    suppose
A8:   x in SCM-Data-Loc;
      reconsider a = x as Element of SCM-Memory by A2,A3;
A9:   s1.a in pi(product(SCM-VAL*SCM-OK),a) by CARD_3:def 6;
      dom(SCM-VAL*SCM-OK) = SCM-Memory by AMI_2:27;
      then
A10:  pi(product(SCM-VAL*SCM-OK),a) = (SCM-VAL*SCM-OK).a by CARD_3:12;
      (SCM*-VAL*SCM+FSA-OK).x = INT by A8,Th5;
      hence s1.x in f.x by A8,A10,A9,AMI_2:8;
    end;
  end;
  then s +* s1 is SCM+FSA-State by A1,A2,PRE_CIRC:6,XBOOLE_1:7;
  hence thesis;
end;
