reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

theorem Th12:
  Px(a,n+2) = 2*a*Px(a,n+1) - Px(a,n) & Py(a,n+2) = 2*a*Py(a,n+1) - Py(a,n)
proof
  set n1=n+1, d=a^2-'1;
  a*a>=0+1 & a*a=a^2 by INT_1:7,SQUARE_1:def 1;
  then
A1: a^2-'1 =a*a-1 by XREAL_1:233;
A2: Px(a,n1) = Px(a,n)*a + Py(a,n)*d & Py(a,n1) = Px(a,n) + Py(a,n)*a
    by HILB10_1:6;
A3:  Px(a,n1+1) = Px(a,n1)*a + Py(a,n1)*d &
  Py(a,n1+1) = Px(a,n1) + Py(a,n1)*a by HILB10_1:6;
A4:a*Px(a,n1) - d*Py(a,n1)  = Px(a,n) by A1,A2;
  a*Py(a,n1) - Px(a,n1)= Py(a,n) by A1,A2;
  hence thesis by A4,A3;
end;
