reserve x,y,P,Q for Integer;
reserve a,b,n for Nat;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d for object;
reserve z for Element of V;
reserve T for TypeSCNominativeData of V,A;
reserve size for non zero Nat;
reserve x0, y0, p0, q0 for Integer;
reserve n0 for Nat;

theorem Th8:
  Lucas_Sequence(a,b,1,-1) = GenFib(a,b)
  proof
    set L = Lucas_Sequence(a,b,1,-1);
    set F = GenFib(a,b);
    dom F = NAT & dom L = NAT by FUNCT_2:def 1;
    hence dom L = dom F;
    let n be object such that
A1: n in dom L;
    defpred P[Nat] means L.$1 = F.$1;
    L.0 = [a,b] by Def3;
    then
A2: P[0] by FIB_NUM3:def 3;
A3: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A4:   P[k];
      thus L.(k+1) = [ (L.k)`2, 1*(L.k)`2 - (-1)*(L.k)`1 ] by Def3
      .= [ (F.k)`2, (F.k)`1 + (F.k)`2 ] by A4
      .= F.(k+1) by FIB_NUM3:def 3;
    end;
    for k being Nat holds P[k] from NAT_1:sch 2(A2,A3);
    hence thesis by A1;
  end;
