reserve n,i,j,k for Nat;
reserve T for TuringStr,
  s for All-State of T;

theorem Th44:
  for T1,T2 being TuringStr, g be Tran-Goal of T2, q be State of
T2, y be Symbol of T2 st g = (the Tran of T2).[q, y] holds (the Tran of T1 ';'
  T2).[ [the AcceptS of T1,q],y] = [[the AcceptS of T1,g`1_3], g`2_3, g`3_3]
proof
  let t1,t2 be TuringStr, g be Tran-Goal of t2,q be State of t2, y be Symbol
  of t2;
  assume
A1: g = (the Tran of t2).[q, y];
  set pF=the AcceptS of t1;
  set x=[[pF,q],y];
  pF in { pF } by TARSKI:def 1;
  then [pF,q] in [: {pF}, the FStates of t2 :] by ZFMISC_1:def 2;
  then
A2: [pF,q] in [: the FStates of t1, {the InitS of t2} :] \/ [: {pF}, the
  FStates of t2 :] by XBOOLE_0:def 3;
  y in (the Symbols of t1) \/ the Symbols of t2 by XBOOLE_0:def 3;
  then reconsider
  xx=x as Element of [: UnionSt(t1,t2), (the Symbols of t1) \/ the
  Symbols of t2 :] by A2,ZFMISC_1:def 2;
A3: SecondTuringState xx =[[pF,q],y]`1`2
    .=q;
A4: SecondTuringSymbol(xx)=[[pF,q],y]`2 by Def28
    .=y;
  thus (the Tran of t1 ';' t2).x = UnionTran(t1,t2).xx by Def31
    .=Uniontran(t1,t2,xx) by Def30
    .=SecondTuringTran(t1,t2,(the Tran of t2).[q,y]) by A3,A4,Def29
    .=[[pF,g`1_3], g`2_3, g`3_3] by A1;
end;
