reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th89:
  for a,b,a1,a2,a3,b1,b2,b3 being Real st a > 0 & b > 0 & a3 >= 0 & b3 >= 0 &
  (a1 > 0 & a2 >= 0 or a1 >= 0 & a2 > 0) &
  (b1 > 0 & b2 >= 0 or b1 >= 0 & b2 > 0) holds
  for n being Nat holds
  (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).n)`1 > 0 &
  (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).n)`2 > 0
  proof
    let a,b,a1,a2,a3,b1,b2,b3 be Real such that
A1: a > 0 & b > 0 and
B1: a3 >= 0 & b3 >= 0 and
A2: (a1 > 0 & a2 >= 0 or a1 >= 0 & a2 > 0) &
    (b1 > 0 & b2 >= 0 or b1 >= 0 & b2 > 0);
    set f = recSeqCart(a,b,a1,a2,a3,b1,b2,b3);
    defpred P[Nat] means (f.$1)`1 > 0 & (f.$1)`2 > 0;
A3: P[0]
    proof
      f.0 = [a,b] by Def10;
      hence thesis by A1;
    end;
A4: P[k] implies P[k+1]
    proof
      f.(k+1) = [a1*(f.k)`1+a2*(f.k)`2+a3,b1*(f.k)`1+b2*(f.k)`2+b3] by Def10;
      hence thesis by B1,A2;
    end;
    P[k] from NAT_1:sch 2(A3,A4);
    hence thesis;
  end;
