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

theorem Th42:
  for s,t being TuringStr, x be Element of UnionSt(s,t) holds ex
  x1 be State of s, x2 be State of t st x=[x1, x2]
proof
  let s,t be TuringStr,x be Element of UnionSt(s,t);
  set q0=the InitS of t, p1=the AcceptS of s,
  A= [: the FStates of s, { q0 } :]
  , B= [: { p1 }, the FStates of t :];
  per cases by XBOOLE_0:def 3;
  suppose
    x in A;
    then consider x1 being State of s, x2 be Element of { q0 } such that
A1: x= [x1, x2] by DOMAIN_1:1;
    take x1,q0;
    thus thesis by A1,TARSKI:def 1;
  end;
  suppose
    x in B;
    then consider x1 being Element of { p1 }, x2 being State of t such that
A2: x= [x1, x2] by DOMAIN_1:1;
    take p1,x2;
    thus thesis by A2,TARSKI:def 1;
  end;
end;
