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

theorem
  EvenFibs (4) = <*1,3*>
proof
  now
    let x be object;
    assume
A1: x in EvenNAT /\ {1,2,3,4};
    then
A2: x in EvenNAT by XBOOLE_0:def 4;
A3: x in {1,2,3,4} by A1,XBOOLE_0:def 4;
    per cases by A3,ENUMSET1:def 2;
    suppose
      x = 2 * (0 qua Nat) + 1;
      hence x in {2,4} by A2,Th51;
    end;
    suppose
      x = 2 * 1;
      hence x in {2,4} by TARSKI:def 2;
    end;
    suppose
      x = 2 * 1 + 1;
      hence x in {2,4} by A2,Th51;
    end;
    suppose
      x = 2 * 2;
      hence x in {2,4} by TARSKI:def 2;
    end;
  end;
  then
A4: EvenNAT /\ {1,2,3,4} c= {2,4};
  set q = {[2,FIB.2],[4,FIB.4]};
  4 in NAT;
  then
A5: 4 in dom FIB by FUNCT_2:def 1;
  reconsider q as FinSubsequence by Th15;
  2 in NAT;
  then
A6: 2 in dom FIB by FUNCT_2:def 1;
A7: FIB | ({2} \/ {4}) = (FIB | {2}) \/ (FIB | {4}) by RELAT_1:78
    .= {[2,FIB.2]} \/ (FIB | {4}) by A6,GRFUNC_1:28
    .= {[2,FIB.2]} \/ {[4,FIB.4]} by A5,GRFUNC_1:28
    .= q by ENUMSET1:1;
  now
    let x be object;
    assume
A8: x in {2,4};
    then x = 2 * 1 or x = 2 * 2 by TARSKI:def 2;
    then
A9: x in EvenNAT;
    x = 2 or x = 4 by A8,TARSKI:def 2;
    then x in {1,2,3,4} by ENUMSET1:def 2;
    hence x in EvenNAT /\ {1,2,3,4} by A9,XBOOLE_0:def 4;
  end;
  then {2,4} c= EvenNAT /\ {1,2,3,4};
  then EvenNAT /\ {1,2,3,4} = {2,4} by A4;
  then EvenFibs (4) = Seq q by A7,ENUMSET1:1,FINSEQ_3:2
    .= <*FIB.2,FIB.4*> by Th16
    .= <*(Fib (2)),FIB.4*> by Def2
    .= <*1,3*> by Def2,Th21,Th23;
  hence thesis;
end;
