reserve J,J1,K for Element of Segm 13,
  b,b1,b2,c,c1,c2 for Element of SCM+FSA-Data-Loc,
  f,f1,f2 for Element of SCM+FSA-Data*-Loc;
reserve k for Nat,
  J,K,L for Element of Segm 13,
  O,P,R for Element of Segm 9;
reserve da for Int-Location,
  fa for FinSeq-Location,
  x,y for set;
reserve la,lb for Nat,
  La for Nat,
  i for Instruction of SCM+FSA,
  I for Instruction of SCM,
  l for Nat,
  LA,LB for Nat,
  dA,dB,dC,dD for Element of SCM+FSA-Data-Loc,
  DA,DB,DC for Element of SCM-Data-Loc,
  fA,fB,fC for Element of SCM+FSA-Data*-Loc,
  f,g for FinSeq-Location,
  A,B for Data-Location,
  a,b,c,db for Int-Location;
reserve S for State of SCM,
  s,s1 for State of SCM+FSA;

theorem Th59:
  Exec(MultBy(a,b), s).IC SCM+FSA = IC s + 1 & Exec(MultBy(a,b),
s).a = s.a * s.b & (for c st c <> a holds Exec(MultBy(a,b), s).c = s.c) & for f
  holds Exec(MultBy(a,b), s).f = s.f
proof
  consider A,B such that
A1: a = A and
A2: b = B and
A3: MultBy(a,b) = MultBy(A,B) by Def9;
  reconsider S = s|SCM-Memory as State of SCM by Th42;
A4: Exec(MultBy(a,b), s)=s +* Exec(MultBy(A,B), S) by A3,Th44;
  hence Exec(MultBy(a,b), s).IC SCM+FSA = Exec(MultBy(A,B), S).IC SCM by Th46
    .= IC S + 1 by AMI_3:5
    .= IC s + 1 by Th55;
  thus Exec(MultBy(a,b), s).a = Exec(MultBy(A,B), S).A by A1,A4,Th47
    .= S.A * S.B by AMI_3:5
    .= S.A * s.b by A2,Th48
    .= s.a * s.b by A1,Th48;
  hereby
    let c such that
A5: c <> a;
    reconsider C = c as Data-Location by Th5;
    thus Exec(MultBy(a,b), s).c = Exec(MultBy(A,B), S).C by A4,Th47
      .= S.C by A1,A5,AMI_3:5
      .= s.c by Th48;
  end;
  let f;
A6: not f in dom Exec(MultBy(A,B), S) by Th37;
  thus Exec(MultBy(a,b), s).f
    = s.f by A4,A6,FUNCT_4:11;
end;
