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

theorem Th58:
  for k being Nat holds FIB | (EvenNAT /\ Seg (2 * k +
  2)) \/ {[2*k+4,FIB.(2 * k + 4)]} = FIB | (EvenNAT /\ Seg (2 * k + 4))
proof
  let k be Nat;
A1: dom FIB = NAT by FUNCT_2:def 1;
  FIB | (EvenNAT /\ Seg (2 * k + 4)) = FIB | ((EvenNAT /\ Seg (2 * k + 2))
  \/ {2 * k + 4}) by Th57
    .= (FIB | (EvenNAT /\ Seg (2 * k + 2))) \/ (FIB |{2 * k + 4}) by RELAT_1:78
    .= FIB | (EvenNAT /\ Seg (2 * k + 2)) \/ {[2*k+4,FIB.(2*k+4)]} by A1,
GRFUNC_1:28;
  hence thesis;
end;
