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

theorem Th61:
  OddFibs (3) = <*1,2*>
proof
  now
    let x be object;
    assume
A1: x in OddNAT /\ {1,2,3};
    then
A2: x in OddNAT by XBOOLE_0:def 4;
A3: x in {1,2,3} by A1,XBOOLE_0:def 4;
    per cases by A3,ENUMSET1:def 1;
    suppose
      x = 2 * (0 qua Nat) + 1;
      hence x in {1,3} by TARSKI:def 2;
    end;
    suppose
      x = 2 * 1;
      hence x in {1,3} by A2,Th52;
    end;
    suppose
      x = 2 * 1 + 1;
      hence x in {1,3} by TARSKI:def 2;
    end;
  end;
  then
A4: OddNAT /\ {1,2,3} c= {1,3};
  set q = {[1,FIB.1],[3,FIB.3]};
  3 in NAT;
  then
A5: 3 in dom FIB by FUNCT_2:def 1;
  reconsider q as FinSubsequence by Th15;
  1 in NAT;
  then
A6: 1 in dom FIB by FUNCT_2:def 1;
A7: FIB | ({1} \/ {3}) = (FIB | {1}) \/ (FIB | {3}) by RELAT_1:78
    .= {[1,FIB.1]} \/ (FIB | {3}) by A6,GRFUNC_1:28
    .= {[1,FIB.1]} \/ {[3,FIB.3]} by A5,GRFUNC_1:28
    .= q by ENUMSET1:1;
  now
    let x be object;
    assume
A8: x in {1,3};
    then x = 2 * (0 qua Nat) + 1 or x = 2 * 1 + 1 by TARSKI:def 2;
    then
A9: x in OddNAT;
    x = 1 or x = 3 by A8,TARSKI:def 2;
    then x in {1,2,3} by ENUMSET1:def 1;
    hence x in OddNAT /\ {1,2,3} by A9,XBOOLE_0:def 4;
  end;
  then {1,3} c= OddNAT /\ {1,2,3};
  then OddNAT /\ {1,2,3} = {1,3} by A4;
  then OddFibs (3) = Seq (FIB | ({1} \/ {3})) by ENUMSET1:1,FINSEQ_3:1
    .= <*FIB.1,FIB.3*> by A7,Th16
    .= <* Fib (1), FIB.3 *> by Def2
    .= <* 1, 2 *> by Def2,Th22,PRE_FF:1;
  hence thesis;
end;
