reserve n, k, r, m, i, j for Nat;

theorem Th57:
  for k being Nat holds (EvenNAT /\ Seg (2 * k + 2)) \/
  {2 * k + 4} = EvenNAT /\ Seg (2 * k + 4)
proof
  let k be Nat;
  2 * k + 4 = 2 * (k + 2);
  then
A1: 2 * k + 4 in EvenNAT;
  2 * k + 3 = 2 * (k + 1) + 1;
  then
A2: {2*k+3} misses EvenNAT by Th51,ZFMISC_1:50;
  EvenNAT /\ Seg (2 * k + 4) = EvenNAT /\ Seg (2 * k + 3 + 1)
    .= EvenNAT /\ (Seg (2 * k + 3) \/ {2 * k + 4}) by FINSEQ_1:9
    .= EvenNAT /\ Seg (2 * k + 3) \/ EvenNAT /\ {2 * k + 4} by XBOOLE_1:23
    .= EvenNAT /\ Seg (2 * k + 2 + 1) \/ {2 * k + 4} by A1,ZFMISC_1:46
    .= EvenNAT /\ (Seg (2 * k + 2) \/ {2 * k + 3}) \/ {2 * k + 4} by FINSEQ_1:9
    .= EvenNAT /\ Seg (2 * k + 2) \/ EvenNAT /\ {2 * k + 3} \/ {2 * k + 4}
  by XBOOLE_1:23
    .= EvenNAT /\ Seg (2 * k + 2) \/ {} \/ {2 * k + 4} by A2
    .= EvenNAT /\ Seg (2 * k + 2) \/ {2 * k + 4};
  hence thesis;
end;
