reserve x,A for set, i,j,k,m,n, l, l1, l2 for Nat;
reserve D for non empty set, z for Nat;
reserve S for COM-Struct;
reserve ins for Element of the InstructionsF of S;
reserve k, m for Nat,
  x, x1, x2, x3, y, y1, y2, y3, X,Y,Z for set;

theorem
  for S being COM-Struct, F, G being Program of S
  holds dom F c= dom (F ';' G)
proof
  let S be COM-Struct, F, G be Program of S;
  set P = F ';' G;
A1: dom P = dom CutLastLoc F \/ dom Reloc(G,card F -' 1)
  by FUNCT_4:def 1;
A2: dom F = dom CutLastLoc F \/ {LastLoc F} by VALUED_1:37;
  let x be object;
  assume
A3: x in dom F;
  per cases by A2,A3,XBOOLE_0:def 3;
  suppose x in dom CutLastLoc F;
    hence thesis by A1,XBOOLE_0:def 3;
  end;
  suppose
A4: x in {LastLoc F};
    then
A5: x = LastLoc F by TARSKI:def 1;
    reconsider f = x as Element of NAT by A4;
A6: f = card F -' 1 by A5,AFINSQ_1:70
      .= card F - 1 + (0 qua Nat) by PRE_CIRC:20;
    card P = card F + card G - 1 by Th11
      .= card F - 1 + card G;
    then f < card P by A6,XREAL_1:6;
    hence thesis by AFINSQ_1:66;
  end;
end;
