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 Th11:
  for S being COM-Struct, F being Program of S,
      G being non empty preProgram of S
  holds card (F ';' G) = card F + card G - 1 &
  card (F ';' G) = card F + card G -' 1
proof
  let S be COM-Struct, F be Program of S, G be non empty preProgram of S;
  set k = card F -' 1;
  dom IncAddr(G,k),dom Reloc(G,k) are_equipotent by VALUED_1:27;
  then
A1: IncAddr(G,k),Reloc(G,k) are_equipotent by PRE_CIRC:21;
  dom CutLastLoc F misses dom Reloc(G,k) by Th9;
  hence card (F ';' G)
  = card CutLastLoc F + card Reloc(G,k) by PRE_CIRC:22
    .= card CutLastLoc F + card IncAddr(G,k) by A1,CARD_1:5
    .= card CutLastLoc F + card dom IncAddr(G,k) by CARD_1:62
    .= card CutLastLoc F + card dom G by Def9
    .= card CutLastLoc F + card G by CARD_1:62
    .= card F - 1 + card G by VALUED_1:38
    .= card F + card G - 1;
  hence thesis by XREAL_0:def 2;
end;
