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

theorem Th43:
  for T1,T2 being TuringStr, g be Tran-Goal of T1,p be State of T1
  , y be Symbol of T1 st p <> the AcceptS of T1 & g = (the Tran of T1).[p, y]
holds (the Tran of T1 ';' T2).[ [p,the InitS of T2],y]
     = [[g`1_3,the InitS of T2]
  , g`2_3, g`3_3]
proof
  let t1,t2 be TuringStr, g be Tran-Goal of t1,p be State of t1, y be Symbol
  of t1;
  assume that
A1: p <> the AcceptS of t1 and
A2: g = (the Tran of t1).[p, y];
  set q0=the InitS of t2;
  set x=[[p,q0],y];
  q0 in { q0 } by TARSKI:def 1;
  then [p,q0] in [: the FStates of t1, {q0} :] by ZFMISC_1:def 2;
  then
A3: [p,q0] in [: the FStates of t1, {q0} :] \/ [: {the AcceptS of t1}, 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 A3,ZFMISC_1:def 2;
A4: FirstTuringState xx = [[p,q0],y]`1`1
    .=[p,q0]`1
    .=p;
A5: FirstTuringSymbol(xx)=[[p,q0],y]`2 by Def27
    .=y;
  thus (the Tran of t1 ';' t2).x = UnionTran(t1,t2).xx by Def31
    .=Uniontran(t1,t2,xx) by Def30
    .=FirstTuringTran(t1,t2,(the Tran of t1).[p,y]) by A1,A4,A5,Def29
    .=[[g`1_3,q0], g`2_3, g`3_3] by A2;
end;
