
theorem Problem50: :: Problem 50
  ex f being Fibonacci-valued sequence of REAL st
     f is increasing with_all_coprime_terms
  proof
    deffunc F(Nat) = Fib primenumber $1;
    consider f being sequence of REAL such that
A1: for n being Nat holds f.n = F(n) from SEQ_1:sch 1;
    for n being Nat holds f.n < f.(n+1)
    proof
      let n be Nat;
B3:   1 < primenumber n by INT_2:def 4;
B1:   f.n = Fib (primenumber n) by A1;
B4:   f.(n+1) = Fib primenumber (n+1) by A1;
      n < n+1 by NAT_1:13; then
      primenumber n < primenumber (n+1) by MOEBIUS2:21;
      hence thesis by B1,B4,B3,FIB_NUM2:46;
    end; then
T1: f is increasing by SEQM_3:def 6;
    for n being Nat holds
      ex fn being Nat st fn = f.n & fn is Fibonacci
    proof
      let n be Nat;
      f.n = Fib (primenumber n) by A1;
      hence thesis;
    end; then
    f is Fibonacci-valued; then
    reconsider f as Fibonacci-valued sequence of REAL;
    for i,j being Nat st i in dom f & j in dom f & i <> j holds
      f.i, f.j are_coprime
    proof
      let i,j be Nat;
      assume
A2:   i in dom f & j in dom f & i <> j;
A3:   f.i = Fib (primenumber i) by A1;
A4:   f.j = Fib (primenumber j) by A1;
      primenumber i <> primenumber j
      proof
        per cases by A2,XXREAL_0:1;
        suppose i < j;
          hence thesis by MOEBIUS2:21;
        end;
        suppose i > j;
          hence thesis by MOEBIUS2:21;
        end;
      end; then
      primenumber i, primenumber j are_coprime by PQCoprime;
      hence thesis by A3,A4,FIB_NUM:5,PRE_FF:1;
    end; then
    f is with_all_coprime_terms;
    hence thesis by T1;
  end;
