reserve p for preProgram of SCM+FSA,
  ic for Instruction of SCM+FSA,
  i,j,k for Nat,
  fa,f for FinSeq-Location,
  a,b,da,db for Int-Location,
  la,lb for Nat;
reserve p1,p2,q for Instruction-Sequence of SCM+FSA;
reserve n for Nat;

theorem Th14:
  for i,k being Nat,p being Program of SCM+FSA,
  s1,s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 &
  (for j holds IC Comput(p1,s1,j) in dom p &
  IC Comput(p2,s2,j) in dom p) &
  Comput(p1,s1,k).IC SCM+FSA = Comput(p2,s2,k).IC
SCM+FSA &
  Comput(p1,s1,k) | (UsedI*Loc p \/ UsedILoc p) =
  Comput(p2,s2,k) | (UsedI*Loc p \/ UsedILoc p) holds
  Comput(p1,s1,i).IC SCM+FSA = Comput(p2,s2,i).IC
SCM+FSA &
  Comput(p1,s1,i) |(UsedI*Loc p \/ UsedILoc p) =
  Comput(p2,s2,i) |(UsedI*Loc p \/ UsedILoc p)
proof
  let i,k be Nat,p be Program of SCM+FSA,s1,s2 be State of SCM+FSA;
  set D= UsedI*Loc p \/ UsedILoc p;
  assume that
A1: k <= i and
A2: p c= p1 and
A3: p c= p2 and
A4: for j holds IC Comput(p1,s1,j) in dom p & IC Comput(
p2,s2,j) in
  dom p and
A5: Comput(p1,s1,k).IC SCM+FSA = Comput(p2,s2,k).IC
SCM+FSA and
A6: Comput(p1,s1,k) | D = Comput(p2,s2,k) | D;
  reconsider t={} as PartState of SCM+FSA by FUNCT_1:104,RELAT_1:171;
  set D1= dom t \/ UsedI*Loc p \/ UsedILoc p;
A7: dom t c= Int-Locations \/ FinSeq-Locations by RELAT_1:38,XBOOLE_1:2;
A8: D1 = D by RELAT_1:38;
  hence Comput(p1,s1,i).IC SCM+FSA = Comput(p2,s2,i).
IC SCM+FSA
  by A1,A2,A3,A4,A5,A6,A7,Th13;
  thus thesis by A1,A2,A3,A4,A5,A6,A7,A8,Th13;
end;
