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 Th15:
  a <= b implies Px(a,n) <= Px(b,n) & Py(a,n) <= Py(b,n)
proof
  assume
A1: a <=b;
  defpred P[Nat] means Px(a,$1) <= Px(b,$1) & Py(a,$1) <= Py(b,$1);
  Px(a,0) = 1 & Py(a,0) = 0 &
  Px(b,0) = 1 & Py(b,0) = 0 by HILB10_1:3;
  then
A2: P[0];
A3: P[k] implies P[k+1]
  proof
    assume
A4:   P[k];
    set k1=k+1;
A5:   a*a =a^2 & b*b=b^2 by SQUARE_1:def 1;
    then a^2>=1+0 & b^2>=1+0 by NAT_1:13;
    then
A6:   a^2-'1 = a^2-1 & b^2-'1 = b^2-1 by XREAL_1:233;
    a^2 <= b^2 by A5,A1,XREAL_1:66;
    then
A7:   a^2-'1 <= b^2-'1 by A6,XREAL_1:9;
    Px(a,k)*a <= Px(b,k)*b & Py(a,k)*(a^2-'1) <= Py(b,k)*(b^2-'1)
      by A1,A7,A4,XREAL_1:66;
    then Px(a,k1)= Px(a,k)*a + Py(a,k)*(a^2-'1) <=
    Px(b,k)*b + Py(b,k)*(b^2-'1) by XREAL_1:7,HILB10_1:6;
    hence Px(a,k1)<=Px(b,k1) by HILB10_1:6;
    Py(a,k)*a <= Py(b,k)*b by A1,A4,XREAL_1:66;
    then Py(a,k1)=Px(a,k) + Py(a,k)*a <= Px(b,k) + Py(b,k)*b
      by A4,XREAL_1:7,HILB10_1:6;
    hence thesis by HILB10_1:6;
  end;
  P[k] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
