reserve a,b,n for Element of NAT;

theorem
  for a,b,n being Element of NAT holds GenFib(a,b,n+2) * GenFib(a,b,n) -
  GenFib(a,b,n+1)|^2 = ((-1)to_power n) * (GenFib(a,b,2)|^2 - GenFib(a,b,1) *
  GenFib(a,b,3))
proof
  let a,b,n be Element of NAT;
  defpred P[Nat] means GenFib(a,b,$1+2)* GenFib(a,b,$1)-GenFib(a,b,$1+1)|^2 =(
  (-1)to_power $1)*(GenFib(a,b,2)|^2-GenFib(a,b,1)*GenFib(a,b,3));
A1: GenFib(a,b,2)=GenFib(a,b,0+2) .=GenFib(a,b,0)+GenFib(a,b,0+1) by Th34
    .=a+GenFib(a,b,1) by Th32
    .=a+b by Th32;
A2: GenFib(a,b,3)=GenFib(a,b,(1+2)) .=GenFib(a,b,1)+GenFib(a,b,1+1) by Th32
    .=b+(a+b) by A1,Th32
    .=2*b+a;
  then
  GenFib(a,b,1+2)* GenFib(a,b,1)-GenFib(a,b,1+1)|^2 =(2*b+a)*b-(a+b)|^2 by A1
,Th32
    .=2*b*b+a*b-((a+b)*(a+b)) by WSIERP_1:1
    .=(-1)*(((a+b)*(a+b))-b*(2*b+a))
    .=((-1)to_power 1)*(((a+b)*(a+b))-b*(2*b+a)) by POWER:25
    .=((-1)to_power 1)*((a+b)|^2-b*GenFib(a,b,3)) by A2,WSIERP_1:1
    .=((-1)to_power 1)*(GenFib(a,b,2)|^2-GenFib(a,b,1)*GenFib(a,b,3)) by A1
,Th32;
  then
A3: P[1];
A4: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume that
A5: P[k] and
A6: P[k+1];
    set d = GenFib(a,b,k+2)*GenFib(a,b,k);
A7: (-1)to_power(k+1)+((-1)to_power k)= ((-1)to_power(k)*(-1)to_power (1)
    )+(-1)to_power k by FIB_NUM2:5
      .=(-1)to_power(k)*(-1)+(-1)to_power k by POWER:25
      .=0;
    GenFib(a,b,2)|^2= GenFib(a,b,0+2)|^2
      .=(GenFib(a,b,0)+GenFib(a,b,0+1))|^2 by Th34
      .=(a+GenFib(a,b,1))|^2 by Th32
      .=(a+b)|^2 by Th32;
    then
A8: GenFib(a,b,2)|^2-GenFib(a,b,1)*GenFib(a,b,3)=(a+b)|^2-b*GenFib(a,b, 1
    +2 ) by Th32
      .=(a+b)|^2-b*(GenFib(a,b,1)+GenFib(a,b,1+1)) by Th34
      .=(a+b)|^2-b*(b+GenFib(a,b,0+2)) by Th32
      .=(a+b)|^2-b*(b+(GenFib(a,b,0)+GenFib(a,b,0+1))) by Th34
      .=(a+b)|^2-b*(b+(a+GenFib(a,b,1))) by Th32
      .=(a+b)|^2-b*(b+(a+b)) by Th32
      .=(a+b)|^2-b*b-b*a-b*b
      .=(a*a+a*b+b*a+b*b)-b*b-b*a-b*b by Th5
      .=a*a+a*b-b*b;
    then d-GenFib(a,b,k+1)|^2 = ((-1)to_power k)*(a*a+a*b-b*b) by A5;
    then
A9: (((-1) to_power (k+1))*(a*a+a*b-b*b))+(d-GenFib(a,b,k+1)|^2) = (((-1)
    to_power (k+1))+((-1)to_power k))*(a*a+a*b-b*b)
      .= (a*a+a*b-b*b)*0 by A7;
    set c = GenFib(a,b,(k+2)+1)*GenFib(a,b,k+1);
A10: c-GenFib(a,b,(k+1)+1)|^2=((-1)to_power (k+1))*(GenFib(a,b,2)|^2 -
    GenFib(a,b,1)*GenFib(a,b,3)) by A6;
A11: c+d-(GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1))|^2 =c+d-(GenFib(a,b,k+1)|^2
    +2*GenFib(a,b,k+1)*GenFib(a,b,(k+1)+1)+ GenFib(a,b,(k+1)+1)|^2) by Th33
      .=((-1)to_power (k+1))*(GenFib(a,b,2)|^2 -GenFib(a,b,1)*GenFib(a,b,3))
    +d-GenFib(a,b,k+1)|^2-2*GenFib(a,b,k+1)* GenFib(a,b,(k+1)+1) by A10
      .=(((-1)to_power (k+1))*(a*a+a*b-b*b)) +d-GenFib(a,b,k+1)|^2-2*GenFib(
    a,b,k+1)*GenFib(a,b,(k+1)+1) by A8
      .=-2*GenFib(a,b,k+1)*GenFib(a,b,(k+1)+1) by A9;
    GenFib(a,b,(k+2)+2)*GenFib(a,b,k+2)-GenFib(a,b,(k+2)+1)|^2 =(GenFib(a
,b,k+2)+GenFib(a,b,(k+2)+1))*GenFib(a,b,k+2)- GenFib(a,b,(k+2)+1)|^2 by Th34
      .=(GenFib(a,b,(k+2)+1)+GenFib(a,b,k+2))*(GenFib(a,b,k)+ GenFib(a,b,k+1
    )) -GenFib(a,b,(k+1)+2)|^2 by Th34
      .=(GenFib(a,b,(k+2)+1)*GenFib(a,b,k)+c+d+GenFib(a,b,k+2)* GenFib(a,b,k
    +1)) -(GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1))|^2 by Th32
      .=GenFib(a,b,(k+2)+1)*GenFib(a,b,k)+GenFib(a,b,k+2)* GenFib(a,b,k+1)+(
    c+d -(GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1))|^2)
      .=GenFib(a,b,(k+2)+1)*GenFib(a,b,k)+GenFib(a,b,k+2)*GenFib(a,b,k+1)+ -
    2*GenFib(a,b,k+1)*GenFib(a,b,k+2) by A11
      .=GenFib(a,b,(k+1)+2)*GenFib(a,b,k)-GenFib(a,b,k+2)*GenFib(a,b,k+1)
      .=(GenFib(a,b,k+1)+GenFib(a,b,(k+1)+1))*GenFib(a,b,k) -GenFib(a,b,k+2)
    *GenFib(a,b,k+1) by Th34
      .=GenFib(a,b,k+1)*GenFib(a,b,k)+d -(GenFib(a,b,k)+GenFib(a,b,k+1))*
    GenFib(a,b,k+1) by Th34
      .=d-(GenFib(a,b,k+1)*GenFib(a,b,k+1))
      .=((-1)to_power k)*1*(GenFib(a,b,2)|^2-GenFib(a,b,1)*GenFib(a,b,3)) by A5
,WSIERP_1:1
      .=((-1)to_power (k)*(1)to_power (2))*(GenFib(a,b,2)|^2 -GenFib(a,b,1)*
    GenFib(a,b,3)) by POWER:26
      .=((-1)to_power (k)*(-1)to_power (2))*(GenFib(a,b,2)|^2 -GenFib(a,b,1)
    *GenFib(a,b,3)) by Th3
      .=((-1)to_power (k+2))*(GenFib(a,b,2)|^2- GenFib(a,b,1)*GenFib(a,b,3))
    by FIB_NUM2:5;
    hence thesis;
  end;
  GenFib(a,b,0+2)* GenFib(a,b,0)-GenFib(a,b,0+1)|^2= GenFib(a,b,2)*a-
  GenFib(a,b,1)|^2 by Th32
    .=(a+b)*a-b|^2 by A1,Th32
    .=a*a+b*a-b*b by WSIERP_1:1
    .=(a+b)*(a+b)-b*(2*b+a)
    .=1*(GenFib(a,b,2)|^2-b*GenFib(a,b,3)) by A1,A2,WSIERP_1:1
    .=((-1)to_power 0)*(GenFib(a,b,2)|^2-b*GenFib(a,b,3)) by POWER:24
    .=((-1)to_power 0)*(GenFib(a,b,2)|^2-GenFib(a,b,1)*GenFib(a,b,3)) by Th32;
  then
A12: P[0];
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A12, A3, A4);
  hence thesis;
end;
