
theorem Th12:
  for F being finite set, A being FinSequence of bool F,
      i being Element of NAT, x,
      J being set holds
    union (Cut (A, i, x), J \ {i}) = union (A,J \ {i})
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT, x, J
  be set;
  thus union (Cut (A, i, x), J\{i}) c= union (A, J\{i})
  proof
    let z be object;
    assume z in union (Cut (A, i, x), J\{i});
    then consider j be set such that
A1: j in J\{i} and
A2: j in dom Cut (A, i, x) and
A3: z in Cut (A, i, x).j by Def1;
    not j in {i} by A1,XBOOLE_0:def 5;
    then i <> j by TARSKI:def 1;
    then
A4: z in A.j by A2,A3,Def2;
    j in dom A by A2,Def2;
    hence thesis by A1,A4,Def1;
  end;
A5: dom Cut (A, i, x) = dom A by Def2;
  thus union (A, J\{i}) c= union (Cut (A, i, x), J\{i})
  proof
    let z be object;
    assume z in union (A, J\{i});
    then consider j be set such that
A6: j in J \ {i} and
A7: j in dom A and
A8: z in A.j by Def1;
    not j in {i} by A6,XBOOLE_0:def 5;
    then i <> j by TARSKI:def 1;
    then Cut (A, i, x).j = A.j by A5,A7,Def2;
    hence thesis by A5,A6,A7,A8,Def1;
  end;
end;
