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

theorem Th55:
  EvenFibs (2) = <*1*>
proof
  now
    let x be object;
    assume
A1: x in EvenNAT /\ {1,2};
    then
A2: x in EvenNAT by XBOOLE_0:def 4;
A3: x in {1,2} by A1,XBOOLE_0:def 4;
    per cases by A3,TARSKI:def 2;
    suppose x = 1;
     then x = 0 + 1;
     then
A4:    x = 2 * (0 qua Nat) + 1;
       ex k being Nat st x = 2*k by A2;
      hence x in {2} by A4;
    end;
    suppose
      x = 2 * 1;
      hence x in {2} by TARSKI:def 1;
    end;
  end;
  then
A5: EvenNAT /\ {1,2} c= {2};
  set q = {[2,FIB.2]};
  reconsider q as FinSubsequence by Th17;
  2 in NAT;
  then
A6: 2 in dom FIB by FUNCT_2:def 1;
  now
    let x be object;
    assume x in {2};
    then x = 2 * 1 by TARSKI:def 1;
    then x in EvenNAT & x in {1,2} by TARSKI:def 2;
    hence x in EvenNAT /\ {1,2} by XBOOLE_0:def 4;
  end;
  then {2} c= EvenNAT /\ {1,2};
  then EvenNAT /\ {1,2} = {2} by A5;
  then EvenFibs (2) = Seq q by A6,FINSEQ_1:2,GRFUNC_1:28
    .= <*FIB.2*> by FINSEQ_3:157
    .= <*1*> by Def2,Th21;
  hence thesis;
end;
