
theorem Th14:
  for F being finite set, A being FinSequence of bool F,
      i being Element of NAT, x,
      J being set st i in dom Cut (A, i, x) & i in J holds
    union (Cut (A, i, x), J) = union (A, J \ {i}) \/ (A.i \ {x})
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT,
      x, J be set such that
A1: i in dom Cut (A, i, x) and
A2: i in J;
  union (Cut (A, i, x), J) = union (Cut (A, i, x), J \ {i}) \/ (Cut (A, i,
  x).i) by A1,A2,Th8
    .= union (A, J \ {i}) \/ (Cut (A, i, x).i) by Th12
    .= union (A, J \ {i}) \/ (A.i \ {x}) by A1,Def2;
  hence thesis;
end;
