reserve a,b,n for Element of NAT;

theorem
  for a,b,k,n being Element of NAT holds GenFib(k*a,k*b,n) = k * GenFib(
  a,b,n )
proof
  let a,b,k,n be Element of NAT;
  defpred P[Nat] means GenFib(k*a,k*b,$1)=k*GenFib(a,b,$1);
A1: for i being Nat st P[i] & P[i+1] holds P[i+2]
  proof
    let i be Nat;
    assume that
A2: P[i] and
A3: P[i+1];
    GenFib(k*a,k*b,i+2)=k*GenFib(a,b,i)+GenFib(k*a,k*b,i+1) by A2,Th34
      .=k*(GenFib(a,b,i)+GenFib(a,b,i+1)) by A3
      .=k*GenFib(a,b,i+2) by Th34;
    hence thesis;
  end;
  GenFib(k*a,k*b,1)=k*b by Th32
    .=k*GenFib(a,b,1) by Th32;
  then
A4: P[1];
  GenFib(k*a,k*b,0)=k*a by Th32
    .=k*GenFib(a,b,0) by Th32;
  then
A5: P[0];
  for i being Nat holds P[i] from FIB_NUM:sch 1 (A5, A4, A1);
  hence thesis;
end;
